Representation theory Lecture Notes: Chapter 6

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 2 December 2013

Reflection groups

Let ${𝔥}^{*}$ be a finite dimensional vector space over $ℂ\text{.}$ The algebra $$S\left({𝔥}^{*}\right),\text{generated by}\hspace{0.17em}x\in {𝔥}^{*}\hspace{0.17em}\text{with relations}\hspace{0.17em}xy=yx\hspace{0.17em}\text{for}\hspace{0.17em}x,y\in {𝔥}^{*}$$ is the symmetric algebra of ${𝔥}^{*}\text{.}$ It can be identified with the algebra of polynomial functions on the dual vector space $𝔥={\text{Hom}}_{ℂ}({𝔥}^{*},ℂ)\text{.}$ As a vector space, $$S\left({𝔥}^{*}\right)=\underset{k\ge 0}{⨁}{S}^k\left({𝔥}^{*}\right),\text{where}{S}^k\left({𝔥}^{*}\right)=ℂ\text{-span}\left\{x_1\cdots x_k\hspace{0.17em}\right|\hspace{0.17em}{x}_1,\dots ,x_k\in {𝔥}^{*}\}\text{.}$$ An element $f\in S\left({𝔥}^{*}\right)$ is homogeneous of degree $k$ if $f\in S^k\left({𝔥}^{*}\right)\text{.}$

The general linear group is the group $GL\left({𝔥}^{*}\right)$ of all invertible linear transformations of ${𝔥}^{*}\text{.}$ A subgroup $W$ of $GL\left({𝔥}^{*}\right)$ acts on $S\left({𝔥}^{*}\right)$ by $$\begin{array}{cc}w\left(p_1{p}_2\right)=\left(w{p}_1\right)\left(w{p}_2\right),\text{for}\hspace{0.17em}w\in W,\hspace{0.17em}{p}_1,p_2\in {𝔥}^{*}\text{.}& \text{(2.1)}\end{array}$$ The invariant ring of $W$ is $$\begin{array}{cc}{S\left({𝔥}^{*}\right)}^W=\{p\in S\left({𝔥}^{*}\right)\hspace{0.17em}|\hspace{0.17em}wp=p\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W\}\text{.}& \text{(2.2)}\end{array}$$

A reflection $s_{\alpha }:{𝔥}^{*}\to {𝔥}^{*}$ is an element of $GL\left({𝔥}^{*}\right)$ such that $\text{codim}\left(\text{ker}(s_{\alpha }-1)\right)=1,$ i.e. $s_{\alpha }$ is diagonalizable and has exactly one eigenvalue not equal to 1. The reflection $s_{\alpha }$ is the reflection in the hyperplane $$\begin{array}{cc}{H}_{\alpha }=\text{ker}(s_{\alpha }-1)={\left({𝔥}^{*}\right)}^{s_{\alpha }}=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}{s}_{\alpha }x=x\}& \text{(2.3)}\end{array}$$ A reflection group is a finite subgroup $W$ of $GL\left({𝔥}^{*}\right)$ generated by reflections, $$W=⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+}⟩,$$ where $R^{+}$ is some index set for the reflections in $W\text{.}$

Let ${𝔥}^{*}$ be a finite dimensional complex vector space and let $W$ be a finite subgroup of $GL\left({𝔥}^{*}\right)\text{.}$ Then $W$ is a reflection group if and only if there are homogeneous elements $e_1,\dots ,e_n\in {S\left({𝔥}^{*}\right)}^W$ such that $$\begin{array}{ccc}ℂ[y_1,\dots ,y_n]& \stackrel{\sim }{⟶}& S\left({𝔥}^{*}\right)\\ y_i& ⟼& e_i,\end{array}$$ is a ring isomorphism.

The theorem says that $W$ is a reflection group if and only if $S\left({𝔥}^{*}\right)$ is generated by $n$ algebraically independent homogeneous polynomials $e_1,\dots ,e_n\text{.}$

Let $W$ be a finite subgroup of $GL\left({𝔥}^{*}\right)$ and let $(,):{𝔥}^{*}\otimes {𝔥}^{*}\to ℂ$ by a bilinear form on ${𝔥}^{*}\text{.}$ Then the bilinear form $⟨,⟩:{𝔥}^{*}\otimes {𝔥}^{*}\to ℂ$ given by $$⟨x_1,x_2⟩=\sum _{w\in W}(w{x}_1,w{x}_2),\text{for}\hspace{0.17em}w\in W,\hspace{0.17em}{x}_1,x_2\in {𝔥}^{*},$$ is $W\text{-invariant,}$ i.e. $$⟨w{x}_1,w{x}_2⟩=⟨x_1,x_2⟩,\text{for all}\hspace{0.17em}{x}_1,x_2\in V,\hspace{0.17em}w\in W\text{.}$$ Thus $W$ is a subgroup of the unitary group $$U({𝔥}^{*},⟨,⟩)=\{g\in GL\left({𝔥}^{*}\right)\hspace{0.17em}|\hspace{0.17em}⟨g{x}_1,g{x}_2⟩=⟨x_1,x_2⟩,\hspace{0.17em}\text{for all}\hspace{0.17em}{x}_1,x_2\in {𝔥}^{*}\}\text{.}$$ The radical of the form $⟨,⟩$ is $$\text{Rad}\left(⟨,⟩\right)=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,y⟩=0\hspace{0.17em}\text{for all}\hspace{0.17em}y\in {𝔥}^{*}\}\text{.}$$ The form $⟨,⟩$ is nondegenerate, if $\text{Rad}\left(⟨,⟩\right)=0\text{.}$ The dual $𝔥={\text{Hom}}_{ℂ}({𝔥}^{*},ℂ)$ of the vector space ${𝔥}^{*}$ has a $W\text{-action}$ given by $$\left(wv\right)\left(x\right)=v\left(w^{-1}x\right),\text{for all}\hspace{0.17em}v\in 𝔥,\hspace{0.17em}x\in {𝔥}^{*}\hspace{0.17em}\text{and}\hspace{0.17em}w\in W,$$ and the map $$\begin{array}{cc}\begin{array}{ccc}{𝔥}^{*}& \stackrel{\sim }{⟶}& 𝔥\\ x& ⟼& ⟨x,·⟩\end{array}& \text{(2.5)}\end{array}$$ is a $W\text{-module}$ homomorphism if and only if the form $⟨,⟩$ is nondegenerate. If ${𝔥}^{*}$ is a simple $W\text{-module}$ then we say that $W$ acts irreducibly on ${𝔥}^{*}\text{.}$ If $⟨,⟩$ is nondegenerate and $W$ acts irreducibly on ${𝔥}^{*}$ then, by Schur’s lemma, the isomorphism in (???) is unique up to constant multiples and so $⟨,⟩$ is, up to constant multiples the unique $W\text{-invariant}$ nondegenerate bilinear form on ${𝔥}^{*}\text{.}$

If $s_{\alpha }\in W$ is a reflection in the hyperplane $H_{\alpha }$ then, for any ${(\alpha \in H_{\alpha })}^{\perp },$ $$H_{\alpha }=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,\alpha ⟩=0\}\text{.}$$ If $\xi \in {ℂ}^{*}$ such that $s_{\alpha }=\xi \alpha$ then, for all $x\in {𝔥}^{*},$ $$\begin{array}{cc}{s}_{\alpha }x=x-⟨x,{\alpha }^{\vee }⟩\alpha ,\text{where}{\alpha }^{\vee }=\frac{(\xi -1)\alpha }{⟨\alpha ,\alpha ⟩}\text{.}& \text{(2.6)}\end{array}$$

Let $W$ be a finite reflection group and let $L$ be a $W\text{-invariant}$ $\mathbb{Z}\text{-lattice}$ in ${𝔥}^{*}\text{.}$ Let $$R=\{\beta \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}\beta \hspace{0.17em}\text{is a minimal length element of}\hspace{0.17em}{H}_{\alpha }^{\perp }\cap L\}\text{.}$$ The set $R$ is the root system of the pair $(W,L)\text{.}$ It is a set of vectors in ${𝔥}^{*}$ which satisfy:

(1)	If, for each $\alpha \in R,$ $$\begin{array}{cccc}{s}_{\alpha }:& {𝔥}^{*}& ⟶& {𝔥}^{*}\\ & x& ⟼& x-⟨x,{\alpha }^{\vee }⟩\alpha \end{array}\text{where}{\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩}$$ is the reflection in the hyperplane $$H_{\alpha }=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩=0\}$$ then $$W=⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R⟩\text{is a finite group.}$$
(2)	The root lattice $$Q=\mathbb{Z}\text{-span}\{\alpha \in R\}$$ is contained in the weight lattice $$P=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩\in \mathbb{Z}\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \in R\}\text{.}$$
(3)	$R$ is $W\text{-invariant,}$ i.e. if $\alpha \in R$ and $w\in W$ then $w\alpha \in R\text{.}$

Let $${𝔥}_{ℝ}^{*}=L{\otimes }_{\mathbb{Z}}ℝ\subseteq {𝔥}^{*}\text{.}$$ The chambers and the alcoves are the connected components of $${𝔥}_{ℝ}^{*}\backslash \left(\bigcup _{\alpha \in ℝ}{H}_{\alpha }\right)\text{and}{𝔥}_{ℝ}^{*}\backslash \left(\bigcup _{\underset{k\in \mathbb{Z}}{\alpha \in ℝ}}{H}_{\alpha ,k}\right),$$ respectively, where $H_{\alpha },$ $\alpha \in R$ and $H_{\alpha ,k},$ $\alpha \in R,$ $k\in \mathbb{Z},$ are the hyperplanes $$H_{\alpha }=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,\alpha ⟩=0\},\text{and}{H}_{\alpha ,k}=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,\alpha ⟩=k\}\text{.}$$ Then

(a)	the reflection $s_{\alpha }:{𝔥}^{*}\to {𝔥}^{*}$ in the hyperplane $H_{\alpha }$ is given by $$s_{\alpha }\left(x\right)=x-⟨x,{\alpha }^{\vee }⟩\alpha ,\text{for}\hspace{0.17em}x\in {𝔥}_{ℝ}^{*},$$
(b)	The reflection $s_{\alpha ,k}:{𝔥}^{*}\to {𝔥}^{*}$ in $H_{\alpha ,k}:{𝔥}_{ℝ}^{*}\to {𝔥}_{ℝ}^{*}$ in $H_{\alpha ,k}$ is given by $$s_{\alpha ,k}\left(x\right)=x-(⟨x,{\alpha }^{\vee }⟩-k)\alpha ,\text{for}\hspace{0.17em}x\in {𝔥}_{ℝ}^{*},$$
(c)	for $\lambda \in P,$ the translation $t_{\lambda }:{𝔥}_{ℝ}^{*}\to {𝔥}_{ℝ}^{*}$ in $\lambda$ is given by $$t_{\lambda }\left(x\right)=x+\lambda ,\text{for}\hspace{0.17em}x\in {𝔥}_{ℝ}^{*}\text{.}$$

For $\alpha \in R,$ $k\in \mathbb{Z}$ and $\lambda \in P,$ $$\begin{array}{cc}{s}_{\alpha ,k}=t_{k\alpha }{s}_{\alpha }=s_{\alpha }{t}_{-k\alpha }\text{and}{s}_{\alpha }{t}_{\lambda }=t_{s_{\alpha }\lambda }{s}_{\alpha }\text{.}& \text{(2.7)}\end{array}$$ The Weyl group the affine Weyl group, and the extended affine Weyl group are, respectively, $$\begin{array}{ccccccc}{W}_0& =& ⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R⟩& =& ⟨w{t}_0\hspace{0.17em}|\hspace{0.17em}w\in W_0⟩& =& W_0⋊\left\{1\right\}\\ \cap |\\ W_Q& =& ⟨s_{\alpha ,k}\hspace{0.17em}|\hspace{0.17em}\alpha \in R,k\in \mathbb{Z}⟩& =& ⟨w{t}_{\lambda }\hspace{0.17em}|\hspace{0.17em}w\in W_0,\lambda \in Q⟩& =& W_0⋊Q\\ \cap |\\ W_P& =& ⟨s_{\alpha ,k}{t}_{\lambda }\hspace{0.17em}|\hspace{0.17em}\alpha \in R,k\in \mathbb{Z},\lambda \in P⟩& =& ⟨w{t}_{\lambda }\hspace{0.17em}|\hspace{0.17em}w\in W_0,\lambda \in P⟩& =& W_0⋊P\end{array}$$ since, by the second relation in (???) $$w{t}_{\lambda }=t_{w\lambda }w,\text{for all}\hspace{0.17em}\lambda \in P,\hspace{0.17em}w\in W_0\text{.}$$

Fix a choice of a chamber $C$ in ${𝔥}_{ℝ}^{*}\text{.}$ The set of positive roots is $$R^{+}=\{\alpha \in R\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩>0,\hspace{0.17em}\text{for all}\hspace{0.17em}x\in C\}\text{and}C=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩>0,\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \in R^{+}\}\text{.}$$ There are bijections $$\begin{array}{ccc}{R}^{+}& \stackrel{1-1}{⟶}& \left\{\text{reflections}\hspace{0.17em}{s}_{\alpha }\hspace{0.17em}\text{in}\hspace{0.17em}W\right\}\\ \alpha & ⟼& s_{\alpha }\end{array}\text{and}\begin{array}{ccc}{W}_0& \stackrel{1-1}{⟶}& \{\text{chambers in}\hspace{0.17em}{𝔥}_{ℝ}^{*}\backslash (\cup H_{\alpha })\}\\ w& ⟼& w^{-1}C\end{array}$$ For $w\in W_0$ the inversion set and the length of $w$ are $$\begin{array}{ccc}R\left(w\right)& =& \{\alpha \in R^{+}\hspace{0.17em}|\hspace{0.17em}{H}_{\alpha }\hspace{0.17em}\text{is between}\hspace{0.17em}C\hspace{0.17em}\text{and}\hspace{0.17em}{w}^{-1}C\}=\{\alpha \in R^{+}\hspace{0.17em}|\hspace{0.17em}w\alpha \notin R^{+}\}\text{and}\\ \ell \left(w\right)& =& \text{Card}\left(R\left(w\right)\right)\text{.}\end{array}$$

Let $H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ be the walls of $C\text{.}$ The corresponding elements ${\alpha }_i$ of $R^{+}$ and the corresponding reflections $s_i=s_{{\alpha }_i}$ are the $$\text{simple roots,}{\alpha }_1,\dots ,{\alpha }_n\text{and the}\text{simple reflections,}{s}_1,\dots ,s_n,$$ respectively. The fundamental weights $${\omega }_1,\dots ,{\omega }_n\in {𝔥}_{ℝ}^{*}\text{are given by}⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},1\le i,j\le n\text{.}$$ Then $$Q=\sum _{i=1}^n\mathbb{Z}{\alpha }_i,P=\sum _{i=1}^n\mathbb{Z}{\omega }_i,\text{and}C=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }_i^{\vee }⟩>0,1\le i\le n\}\text{.}$$

Let $A^{+}$ be the unique alcove contained in $C$ whose closure contains $0\text{.}$ The element $\phi \in R$ such that $H_{\phi ,1}$ is a wall of $A^{+}$ is the highest root of $R\text{.}$ Then $$A^{+}=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}x\in C\hspace{0.17em}\text{and}\hspace{0.17em}⟨x,{\phi }^{\vee }⟩<1\}\text{and}\begin{array}{ccc}{W}_Q& \stackrel{1-1}{⟷}& \left\{\text{alcoves}\right\}\\ w& ⟷& w^{-1}{A}^{+}\end{array}$$ is a bijection. The walls of $A^{+}$ are $H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ and $H_{{\alpha }_0}=H_{\varphi ,1}$ and the reflection in $H_{{\alpha }_0}$ is $$\begin{array}{cc}{s}_0=t_{\phi }{s}_{\phi }=s_{\phi }{t}_{-\phi }\text{.}& \text{(2.8)}\end{array}$$ For $w\in W_P$ the inversion set and the length of $w$ are given by $$\begin{array}{cc}R\left(w\right)=\left\{H_{\alpha ,k}\hspace{0.17em}\right|\hspace{0.17em}{H}_{\alpha ,k}\hspace{0.17em}\text{is between}\hspace{0.17em}{A}^{+}\hspace{0.17em}\text{and}\hspace{0.17em}w{A}^{+}\},\text{and}\ell \left(w\right)=\text{Card}\left(R\left(w\right)\right)\text{.}& \text{(2.9)}\end{array}$$ If $w\in W_0$ and $\lambda \in P$ then $$\begin{array}{cc}\ell \left(w{t}_{\lambda }\right)=\sum _{\underset{H_{\alpha }\in R\left(w\right)}{\alpha \in R^{+}}}|⟨\lambda ,{\alpha }^{\vee }⟩+1|+\sum _{\underset{H_{\alpha }\notin R\left(w\right)}{\alpha \in R^{+}}}\left|⟨\lambda ,{\alpha }^{\vee }⟩\right|\text{.}& \text{(2.10)}\end{array}$$ Define $$\begin{array}{cc}\Omega =\{g\in W_P\hspace{0.17em}|\hspace{0.17em}\ell \left(g\right)=0\}\text{.}& \text{(2.11)}\end{array}$$ If $g\in \Omega$ then $g{A}^{+}=A^{+}$ and so $g$ induces a permutation (also denoted by $g\text{)}$ of the walls $H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ of $A^{+}\text{.}$ Then $$g{s}_i{g}^{-1}=s_{g\left(i\right)},\text{for}\hspace{0.17em}g\in \Omega ,\hspace{0.17em}0\le i\le n\text{.}$$ Thus $\Omega$ corresponds to automorphisms of the Dynkin diagram.

The basic data

A reflection in $O\left({ℝ}^n\right)$ is an element $s$ such that $\text{codim}\left(\text{ker}(s-1)\right)=1\text{.}$ A finite real reflection group is a finite subgroup $W$ of $GL\left({ℝ}^n\right)$ which is generated by reflections. Any nondegenerate symmetric bilinear form $(,):{ℝ}^n\times {ℝ}^n\to ℝ$ can be symmetrized to produce a form $⟨,⟩:{ℝ}^n\times {ℝ}^n\to ℝ$ given by $$⟨x,y⟩=\sum _{w\in W}(wx,wy),\text{for}\hspace{0.17em}x,y\in {ℝ}^n,$$ which is $W\text{-invariant,}$ $$⟨wx,wy⟩=⟨x,y⟩,\text{for all}\hspace{0.17em}x,y\in ℝ,\hspace{0.17em}w\in W\text{.}$$ Let $R^{+}$ be an index set for the reflections in $W\text{.}$ Each reflection $s_{\alpha },$ $\alpha \in R^{+},$ in $W$ fixes a hyperplane $$H_{\alpha }=\{x\in {ℝ}^n\hspace{0.17em}|\hspace{0.17em}{s}_{\alpha }x=x\}\text{.}$$ For any fixed choice of $\alpha \in {ℝ}^n$ with $\alpha \perp H_{\alpha }$ (i.e. $⟨\alpha ,x⟩=0$ for all $x\in H_{\alpha }\text{)}$ the reflection $s_{\alpha }:{ℝ}^n\to {ℝ}^n$ is given by $$s_{\alpha }y=y-⟨y,{\alpha }^{\vee }⟩\alpha ,\text{where}{\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩},$$ since the linear transformation defined by the right hand side of the formula acts as the identity on ${\left(H_{\alpha }\right)}^{\perp }$ and takes $\alpha$ to $-\alpha \text{.}$ The hyperplanes $H_{\alpha },$ $\alpha \in {ℝ}^n,$ cut the whole space ${ℝ}^n$ into a collection of cones, each of which is a fundamental chamber for the action of $W\text{.}$

The basic data is $W$ a finite real reflection group,
$C$ a fixed fundamental chamber for $W,$
$L$ a $W\text{-invariant}$ lattice. For each reflecting hyperplane $H_{\alpha },$ fix $\alpha \in {ℝ}^n$ so that $⟨\alpha ,\alpha ⟩$ is minimal such that $$\alpha \in L,\alpha \perp H_{\alpha },\text{and}⟨\alpha ,x⟩>0,\text{for all}\hspace{0.17em}x\in C,$$ For $\lambda \in {ℝ}^n$ let $$⟨\lambda ,{\alpha }^{\vee }⟩=\text{the distance from}\hspace{0.17em}\lambda \hspace{0.17em}\text{to the hyperplane}\hspace{0.17em}{H}_{\alpha },$$ with the positive side of $H_{\alpha }$ being the side towards the chamber $C$ so that $$C=\{x\in {ℝ}^n\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩>0\hspace{0.17em}\text{for all}\hspace{0.17em}{s}_{\alpha }\}\text{.}$$ For each reflection $s_{\alpha }$ in $W$ let $\alpha \in P$ be such that $$H_{\alpha }=\{x\in R^n\hspace{0.17em}|\hspace{0.17em}⟨x,\alpha ⟩=0\},$$ $\alpha$ is on the positive side of $H_{\alpha }$ and and $⟨\alpha ,\alpha ⟩$ is minimal. These are the positive roots $\alpha \in R^{+}\text{.}$

Let $$P=\sum _{i\in I}\mathbb{Z}{\omega }_i,P^{+}=\sum _{i\in I}{\mathbb{Z}}_{\ge 0}{\omega }_i,P^{++}=\sum _{i\in I}{\mathbb{Z}}_{>0}{\omega }_i\text{.}$$ Let $$\rho =\sum _{i\in I}{\omega }_i\text{so that}\begin{array}{ccc}{P}^{+}& \stackrel{\sim }{⟶}& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}\text{is a bijection.}$$ The raising operators are $$\begin{array}{cccc}{R}_{\alpha }:& P& ⟶& P\\ & \gamma & ⟼& \gamma +\alpha \end{array}$$ and the dominance order is the partial order on $P$ given by $$\lambda \ge \mu \text{if}\lambda =R_{b_1}\cdots R_{{\beta }_k}\mu \text{for some sequence}\hspace{0.17em}{\beta }_1,\dots ,{\beta }_k\in R^{+}\text{.}$$ The dominance order on $W$ is the partial order defined by $$v\le w\text{if}v\rho \le w\rho \text{.}$$ This partial order on $W$ is often called the Bruhat order or the Bruhat-Chevalley order.

A fundamental theorem is the following theorem of Chevalley.

The data $(W,C,L)$ is equivalent to a different data $(G\subseteq B\subseteq T)$ where $G$ is a connected reductive algebraic group over $ℂ,$
$B$ is a Borel subgroup of $G,$ and
$T$ is a maximal torus of $G\text{.}$ These data are equivalent in the sense that each can be uniquely reconstructed from the other.

Example. The basic data corresponding to the group ${GL}_n\left(ℂ\right)\text{.}$

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be an orthonormal basis of $$\begin{array}{ccc}{ℝ}^n& =& \sum _{i-1}^nℝ{\epsilon }_i,\\ W& =& S_n,\text{acting on}\hspace{0.17em}{ℝ}^n\hspace{0.17em}\text{by permuting the}\hspace{0.17em}{\epsilon }_i,\\ C& =& \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_1>{\mu }_2>\cdots >{\mu }_n\},\\ L& =& \sum _{i=1}^n\mathbb{Z}{\epsilon }_i,\end{array}$$ and $$\begin{array}{ccc}{R}^{+}& =& \{{\epsilon }_i-{\epsilon }_j\hspace{0.17em}|\hspace{0.17em}1\le i<j\le n\},\\ {\alpha }_i& =& {\epsilon }_i-{\epsilon }_{i+1},\text{for}\hspace{0.17em}1\le i\le n-1,\\ s_{{\epsilon }_i-{\epsilon }_j}& =& (i,j),\text{the transposition switching}\hspace{0.17em}i\hspace{0.17em}\text{and}\hspace{0.17em}j\hspace{0.17em}\text{in}\hspace{0.17em}{S}_n,\\ s_i& =& (i,i+1),1\le i\le n-1\text{.}\end{array}$$

Symmetric functions

The group algebra of $P$ is $$\mathbb{Z}\left[P\right]=\mathbb{Z}\text{-span}\left\{e^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in P\}\text{with}{e}^{\mu }{e}^{\nu }=e^{\mu +\nu },$$ for all $\mu ,\nu \in P\text{.}$ Define $${\mathbb{Z}\left[P\right]}^W=\{f\in \mathbb{Z}\left[P\right]\hspace{0.17em}|\hspace{0.17em}wf=f\}\text{and}A=\{f\in \mathbb{Z}\left[P\right]\hspace{0.17em}|\hspace{0.17em}wf={(-1)}^{\ell \left(w\right)}f\}\text{.}$$ Define $$a_{\mu }=\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{e}^{w\mu },\text{for every}\hspace{0.17em}\mu \in P\text{.}$$ The elements $a_{\mu }\in A$ satisfy

(a)	$a_{w\mu }={(-1)}^{\ell \left(w\right)}{a}_{\mu },$
(b)	$a_{\mu }=0$ if $⟨\mu ,{\alpha }_i^{\vee }⟩=0\text{.}$
(c)	$a_{\rho }={\prod }_{{\alpha }_{\ell }\in R^{+}}(e^{\alpha /2}-e^{-\alpha /2})={\sum }_{w\in W}{(-1)}^{\ell \left(w\right)}{e}^{w\rho },$
(d)	$a_{\mu }$ is divisible by $a_{\rho }\text{.}$

The orbit sums $$m_{\lambda }=\sum _{\gamma \in W\lambda }{e}^{\gamma },\lambda \in P^{+},\text{and the}{a}_{\mu }=\sum _{w\in W}{\left(-1\right)}^{\ell \left(w\right)}{e}^{w\mu },\mu \in P^{++},$$ form bases of ${\mathbb{Z}\left[P\right]}^W$ and $A,$ respectively. The Weyl characters are $$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }}=\frac{{\displaystyle \sum _{w\in W}}{(-1)}^{\ell \left(w\right)}{e}^{w(\lambda +\rho )}}{{\displaystyle \sum _{w\in W}}{(-1)}^{\ell \left(w\right)}{e}^{w\rho }},\lambda \in P^{+}\text{;}$$ the inverse images of the elements of our favorite basis $\left\{a_{\lambda +\rho }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ of $A$ under the vector space isomorphism $$\begin{array}{ccc}{\mathbb{Z}\left[P\right]}^W& \stackrel{\sim }{⟶}& A\\ f& ⟼& f{a}_{\rho }\\ g/a_{\rho }& ⟵& g\\ s_{\lambda }& ⟷& a_{\lambda +\rho }\end{array}$$ (a vector space generalization of the bijection in (???)). Thus the $s_{\lambda },$ $\lambda \in P^{+},$ are a basis of ${\mathbb{Z}\left[P\right]}^W\text{.}$

Let $f={\sum }_{\mu \in P}{f}_{\mu }{e}^{\mu }\in {\mathbb{Z}\left[P\right]}^W\text{.}$ Then $$f=\sum _{\lambda \in P^{+}}{\eta }_{\lambda }{s}_{\lambda }\text{where}{\eta }_{\lambda }=\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{f}_{\mu +\rho -w\rho }\text{.}$$

		Proof.
	$$\begin{array}{ccc}{\eta }_{\lambda }& =& \left(\text{coefficient of}\hspace{0.17em}{s}_{\lambda }\hspace{0.17em}\text{in}\hspace{0.17em}f\right)=\left(\text{coefficient of}\hspace{0.17em}{a}_{\lambda +\rho }\hspace{0.17em}\text{in}\hspace{0.17em}f{a}_{\rho }\right)\\ & =& \left(\text{coefficient of}\hspace{0.17em}{e}^{\lambda +\rho }\hspace{0.17em}\text{in}\hspace{0.17em}\sum _{\mu \in P}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{f}_{\mu }{e}^{\mu +w\rho }\right)\text{.}\end{array}$$ $\square$

For $\lambda \in P^{+},$ $\mu \in P$ define $K_{\lambda \mu }$ by $$s_{\lambda }=\sum _{\mu \in P}{K}_{\lambda \mu }{e}^{\mu }=\sum _{\mu \in P^{+}}{K}_{\lambda \mu }{m}_{\mu }\text{.}$$ Then $$K_{\lambda \lambda }=1,K_{\lambda ,w\mu }=K_{\lambda \mu }\text{and}{K}_{\lambda \mu }=0\text{unless}\mu \le \lambda \text{.}$$

		Proof.
	By (???) the coefficients $K_{\lambda \mu }^{-1}$ defined by $$m_{\mu }=\sum _{\lambda \in P^{+}}{s}_{\lambda }{K}_{\lambda \mu }^{-1}\text{are given by}{K}_{\lambda \mu }^{-1}=\sum _{w\in W}{(-1)}^{\ell \left(w\right)}\delta (\mu +\rho -w\rho \in W\lambda ),$$ where $\delta (\nu \in W\lambda )=\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}\nu \in W\lambda ,\\ 0,& \text{otherwise.}\end{array}$ $\square$

If $\nu \in {𝔥}_{ℝ}^{*}$ and $f\in {\displaystyle \sum _{\mu \in P}}{f}_{\mu }{e}^{\nu }\in \mathbb{Z}\left[P\right]$ define $f\left(e^{\nu }\right)={\displaystyle \sum _{\mu \in P}}{f}_{\mu }{e}^{⟨\mu ,\nu ⟩}\text{.}$ Let $\lambda \in P^{+}$ and let $q=e^t\text{.}$ Then $$s_{\lambda }\left(q^{\rho }\right)=\prod _{\alpha \in R^{+}}\frac{\left[⟨\lambda +\rho ,{\alpha }^{\vee }⟩\right]}{\left[⟨\rho ,{\alpha }^{\vee }⟩\right]}\text{and}{s}_{\lambda }\left(1\right)=\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}$$ where $\left[k\right]=(q^k-1)/(q-1)$ for an integer $k\ne 0\text{.}$

		Proof.
	$$\begin{array}{ccc}{s}_{\lambda }\left(q^{\rho }\right)=s_{\lambda }\left(e^{t\rho }\right)& =& \frac{a_{\lambda +\rho }\left(e^{t\rho }\right)}{a_{\rho }\left(e^{t\rho }\right)}=\frac{{\displaystyle \sum _{w\in W}}{(-1)}^{\ell \left(w\right)}{e}^{⟨w(\lambda +\rho ),t\rho ⟩}}{a_{\rho }\left(e^{t\rho }\right)}=\frac{{\displaystyle \sum _{w\in W}}{(-1)}^{\ell \left(w\right)}{e}^{⟨t\rho ,w(\lambda +\rho )⟩}}{a_{\rho }\left(e^{t\rho }\right)}\\ & =& \frac{a_{\rho }\left(e^{t(\lambda +\rho )}\right)}{a_{\rho }\left(e^{t\rho }\right)}=\prod _{\alpha \in R^{+}}\frac{e^{⟨\alpha /2,t(\lambda +\rho )⟩}-e^{⟨-\alpha /2,t(\lambda +\rho )⟩}}{e^{⟨\alpha /2,t\rho ⟩}-e^{⟨-\alpha /2,t\rho ⟩}}\\ & =& \prod _{\alpha \in R^{+}}\frac{e^{t⟨\lambda +\rho ,\alpha /2⟩}-e^{-t⟨\lambda +\rho ,\alpha /2⟩}}{e^{t⟨\rho ,\alpha /2⟩}-e^{-t⟨\rho ,\alpha /2⟩}}\end{array}$$ For part (b) $$s_{\lambda }\left(1\right)=s_{\lambda }\left(e^0\right)=\underset{t\to 0}{\text{lim}}{s}_{\lambda }\left(e^{t\rho }\right)=\underset{t\to 0}{\text{lim}}\prod _{\alpha \in R^{+}}\frac{e^{t⟨\lambda +\rho ,\alpha /2⟩}-e^{-t⟨\lambda +\rho ,\alpha /2⟩}}{e^{t⟨\rho ,\alpha /2⟩}-e^{-t⟨\rho ,\alpha /2⟩}}=\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,\alpha /2⟩}{⟨\rho ,\alpha /2⟩}$$ $\square$

$\mathbb{Z}\left[P\right]$ is a free ${\mathbb{Z}\left[P\right]}^W\text{-module}$ with basis $$\left\{e^{{\lambda }_w}\hspace{0.17em}\right|\hspace{0.17em}w\in W\}\text{where}{\lambda }_w=w^{-1}\left(\sum _{s_{i}w<w}{\omega }_i\right)\text{.}$$

		Proof.
	$\square$

For each $i\in I$ define $${\Delta }_i:\mathbb{Z}\left[P\right]⟶\mathbb{Z}\left[P\right]\text{by}{\Delta }_{i}f=\frac{e^{{\alpha }_i}f-s_{i}f}{e^{{\alpha }_i}-1}$$ ????is this formula for ${\Delta }_i$ correct???? and for each $\mu \in P$ define $$X^{\lambda }:\mathbb{Z}\left[P\right]⟶\mathbb{Z}\left[P\right]\text{by}{X}^{\lambda }f=e^{\lambda }f\text{.}$$ The operators ${\Delta }_i$ and $X^{\lambda }$ satisfy the following relations $$\begin{array}{cccc}\underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{{\Delta }_i{\Delta }_j{\Delta }_i\cdots }& =& \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{{\Delta }_j{\Delta }_i{\Delta }_j\cdots }& \text{for all}\hspace{0.17em}i\ne j,\\ {\Delta }_i^2& =& {\Delta }_i,& \text{for all}\hspace{0.17em}i\in I,\\ X^{\lambda }{\Delta }_i& =& {\Delta }_i{X}^{s_i\lambda }+\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},& \text{for all}\hspace{0.17em}i\in I,\hspace{0.17em}\lambda \in P\text{.}\end{array}$$

		Proof.
	For the second relation: $s_i{\Delta }_i\left(f\right)={\Delta }_i\left(f\right)$ and ${\Delta }_i\left(g\right)=g$ if $s_{i}g=g\text{.}$ So ${\Delta }_i^2={\Delta }_i\text{.}$ $\square$

The affine Hecke algebra $\stackrel{\sim }{H}$

Fix $q\in {ℂ}^{*}\text{.}$ The affine Hecke algebra $\stackrel{\sim }{H}$ is the algebra over $ℂ$ given by generators $T_i,$ $i\in I,$ and $X^{\lambda },$ $\lambda \in P,$ and relations $$\begin{array}{cccc}\underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{T_i{T}_j{T}_i\cdots }& =& \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{T_j{T}_i{T}_j\cdots }& \text{for all}\hspace{0.17em}i\ne j,\\ T_i^2& =& (q-q^{-1})T_i+1,& \text{for all}\hspace{0.17em}i\in I,\\ X^{\lambda }{T}_i& =& T_i{X}^{s_i\lambda }+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},& \text{for all}\hspace{0.17em}i\in I,\hspace{0.17em}\lambda \in P\text{.}\end{array}$$

The affine Hecke algebra $\stackrel{\sim }{H}$ is the algebra given by generators $T_w,$ $w\in \stackrel{\sim }{W},$ and relations $$\begin{array}{c}{T}_{w_1}{T}_{w_2}=T_{w_1{w}_2},\text{if}\hspace{0.17em}\ell \left(w_1{w}_2\right)=\ell \left(w_1\right)+\ell \left(w_2\right),\\ T_{s_i}{T}_w=(q-q^{-1})T_w+T_{s_{i}w},\text{if}\hspace{0.17em}\ell \left(s_{i}w\right)<\ell \left(w\right)\text{.}\end{array}$$

The conversion between the two presentations is given by $$X^{\lambda }=T_{t_{\mu }}{T}_{\nu }^{-1},\text{if}\hspace{0.17em}\lambda =\mu -\nu \hspace{0.17em}\text{with}\hspace{0.17em}\mu ,\nu \hspace{0.17em}\text{dominant,}$$

The algebra $\stackrel{\sim }{H}$ has bases $$\left\{X^{\lambda }{T}_w\hspace{0.17em}\right|\hspace{0.17em}w\in W_0,\lambda \in L\}\text{and}\left\{T_w{X}^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}w\in W_0,\lambda \in L\}\text{.}$$

Let $$1_0=\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\text{and}{\epsilon }_0=\sum _{w\in W}{(-q)}^{-\ell \left(w\right)}{T}_w\text{.}$$ For each $\mu \in P$ define $$A_{\mu }={\epsilon }_0{X}^{\mu }{1}_0,\text{and}{P}_{\lambda }=\frac{q^{\ell \left(w_0\right)}}{W_0\left(q^2\right)}\sum _{w\in W}w\left(X^{\lambda }\prod _{\alpha \in R^{+}}\frac{q-q^{-1}{X}^{-\alpha }}{1-X^{-\alpha }}\right)$$

Let $Z\left(\stackrel{\sim }{H}\right)$ be the center of $\stackrel{\sim }{H}\text{.}$ Then $$\begin{array}{ccccccc}Z\left(\stackrel{\sim }{H}\right)& =& {ℂ\left[P\right]}^W& \stackrel{\sim }{⟶}& 1_0\stackrel{\sim }{H}{1}_0& \stackrel{\sim }{⟶}& {\epsilon }_0\stackrel{\sim }{H}{1}_0\\ & & f& ⟼& f{1}_0& ⟼& A_{\rho }f{1}_0\\ & & s_{\lambda }& ⟷& s_{\lambda }{1}_0& ⟷& A_{\lambda +\rho }\\ & & P_{\lambda }& ⟷& 1_0{X}^{\lambda }{1}_0\end{array}$$ where the first map is a ring isomorphism and the second is a vector space isomorphism. $$\left\{A_{\lambda +\rho }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}\hspace{0.17em}\text{is a basis of}\hspace{0.17em}{\epsilon }_0\stackrel{\sim }{H}{1}_0\text{and}\left\{1_0{X}^{\lambda }{1}_0\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}\hspace{0.17em}\text{is a basis of}\hspace{0.17em}{1}_0\stackrel{\sim }{H}{1}_0,$$ and the corresponding bases of $Z\left(\stackrel{\sim }{H}\right)={ℂ\left[P\right]}^W$ are $$\left\{s_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}\text{and}\left\{P_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}\text{.}$$

		Proof.
	(a) First show that $Z\left(\stackrel{\sim }{H}\right)={ℂ\left[P\right]}^W\text{.}$ Assume $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w\in Z\left(\stackrel{\sim }{H}\right)\text{.}$$ Let $m\in W$ be maximal in Bruhat order subject to $c_{\gamma ,m}\ne 0$ for some $\gamma \in L\text{.}$ If $m\ne 1$ there exists a dominant $\mu \in L$ such that $c_{\gamma +\mu -m\mu ,m}=0$ (otherwise $c_{\gamma +\mu -m\mu ,m}\ne 0$ for every dominant $\mu \in L,$ which is impossible since $z$ is a finite linear combination of $X^{\lambda }{T}_w\text{).}$ Since $z\in Z\left(\stackrel{\sim }{H}\right)$ we have $$z=X^{-\mu }z{X}^{\mu }=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda -\mu }{T}_w{X}^{\mu }\text{.}$$ Repeated use of the relation (4.11) yields $$T_w{X}^{\mu }=\sum _{\nu \in L,v\in W}{d}_{\nu ,v}{X}^{\nu }{T}_v$$ where $d_{\nu ,v}$ are constants such that $d_{w\mu ,w}=1,$ $d_{\nu ,w}=0$ for $\nu \ne w\mu ,$ and $d_{\nu ,v}=0$ unless $v\le w\text{.}$ So $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w=\sum _{\lambda \in L,w\in W}\sum _{\nu \in L,v\in W}{c}_{\lambda ,w}{d}_{\nu ,v}{X}^{\lambda -\mu +\nu }{T}_v$$ and comparing the coefficients of $X^{\gamma }{T}_m$ gives $c_{\gamma ,m}=c_{\gamma +\mu -m\mu ,m}{d}_{m\mu ,m}\text{.}$ Since $c_{\gamma +\mu -m\mu ,m}=0$ it follows that $c_{\gamma ,m}=0,$ which is a contradiction. Hence $z\in {\sum }_{\lambda \in L}{c}_{\lambda }{X}^{\lambda }\in ℂ\left[X\right]\text{.}$ The relation (4.11) gives $$z{T}_i=T_{i}z=\left(s_{i}z\right)T_i+(q-q^{-1})z\prime$$ where $z\prime \in ℂ\left[X\right]\text{.}$ Comparing coefficients of $X^{\lambda }$ on both sides yields $z\prime =0\text{.}$ Hence $z{T}_i=\left(s_{i}z\right)T_i,$ and therefore $z=s_{i}z$ for $1\le i\le n\text{.}$ So $z\in {ℂ\left[X\right]}^W\text{.}$ (b) Since $X^{\lambda }+X^{s_i\lambda }$ is in the center of the tiny little affine Hecke algebra generated by $T_i$ and the $X^{\lambda },$ $$\begin{array}{ccc}{A}_{\lambda }+A_{s_i\lambda }& =& {\epsilon }_0(X^{\lambda }+X^{s_i\lambda })1_0=q^{-1}{\epsilon }_0(X^{\lambda }+X^{s_i\lambda })T_i{1}_0\\ & =& q^{-1}{\epsilon }_0{T}_i(X^{\lambda }+X^{s_i\lambda })1_0=-q^{-2}{\epsilon }_0(X^{\lambda }+X^{s_i\lambda })1_0=-q^{-2}(A_{\lambda }+A_{s_i\lambda })\text{.}\end{array}$$ Thus it follows that $A_{\lambda }+A_{s_i\lambda }=0$ and so $$A_{w\lambda }={(-1)}^{\ell \left(w\right)}{A}_{\lambda },\text{for all}\hspace{0.17em}\lambda \in P,\hspace{0.17em}w\in W\text{.}$$ Thus $\left\{A_{\lambda +\rho }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ is a basis of ${\epsilon }_0\stackrel{\sim }{H}{1}_0$ Since $${\epsilon }_0{a}_{\lambda }{1}_0={\epsilon }_0\left(\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{X}^{w\lambda }\right)1_0=\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{A}_{w\lambda }=\left|W\right|A_{\lambda },$$ and $s_{\lambda }\in Z\left(\stackrel{\sim }{H}\right),$ $$A_{\rho }{s}_{\lambda }{1}_0=\frac{1}{\left|W\right|}{\epsilon }_0{a}_{\rho }{1}_0{s}_{\lambda }{1}_0=\frac{1}{\left|W\right|}{\epsilon }_0{a}_{\rho }{s}_{\lambda }{1}_0^2=\frac{1}{\left|W\right|}{\epsilon }_0{a}_{\lambda +\rho }{1}_0=A_{\lambda +\rho }\text{.}$$ Since $\left\{s_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ is a basis of ${ℂ\left[P\right]}^W$ and $\left\{A_{\lambda +\rho }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ is a basis of ${\epsilon }_0\stackrel{\sim }{H}{1}_0$ the composite map $$\begin{array}{ccccccc}Z\left(\stackrel{\sim }{H}\right)& \stackrel{1_0}{⟶}& Z\left(\stackrel{\sim }{H}\right)1_0& \hookrightarrow & 1_0\stackrel{\sim }{H}{1}_0& \stackrel{A_{\rho }}{⟶}& {\epsilon }_0\stackrel{\sim }{H}{1}_0\\ f& ⟼& f{1}_0& ⟼& f{1}_0& ⟼& A_{\rho }f{1}_0\\ s_{\lambda }& ⟼& s_{\lambda }{1}_0& ⟼& s_{\lambda }{1}_0& ⟼& A_{\lambda +\rho }\end{array}$$ is a vector space isomorphism. (c) Now show that $P_{\lambda }{1}_0=1_0{X}^{\lambda }{1}_0\text{.}$ First we can do a rank 1 calculation: $$\begin{array}{ccc}\left(q^{-1}{T}_i\right)X^{\lambda }{U}^{+}& =& (q^{-1}{X}^{\lambda }+X^{s_i\lambda }{T}_i+(q-q^{-1}))\left(\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}\right)U^{+}\\ & =& \frac{1}{1-X^{-{\alpha }_i}}\left(\begin{array}{c}{q}^{-1}{X}^{\lambda }-q^{-1}{X}^{\lambda -{\alpha }_i}+q{X}^{s_i\lambda }-q{X}^{s_i\lambda -{\alpha }_i}\\ +q{X}^{\lambda }-q{X}^{s_i\lambda }-q^{-1}{X}^{\lambda }+q^{-1}{X}^{s_i\lambda }\end{array}\right)U^{+}\\ & =& {(1-X^{-{\alpha }_i})}^{-1}(-q^{-1}{X}^{\lambda -{\alpha }_i}-q{X}^{s_i\lambda -{\alpha }_i}+q{X}^{\lambda }+q^{-1}{X}^{s_i\lambda })U^{+}\\ & =& {(1-X^{-{\alpha }_i})}^{-1}(X^{\lambda }(q-q^{-1}{X}^{-{\alpha }_i})+X^{s_i\lambda }(q^{-1}-q{X}^{-{\alpha }_i}))U^{+}\\ & =& (1+s_i)\left(\frac{q-q^{-1}{X}^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}{X}^{\lambda }\right)U^{+}\end{array}$$ Now $U^{+}$ can be written as a linear combination of products of $q^{-1}+T_i$ and so $U^{+}{X}^{\lambda }{U}^{+}$ can be written as a linear combination of terms of the form $$(1+s_{i_1})\left(\frac{q-q^{-1}{X}^{-{\alpha }_{i_1}}}{1-X^{-{\alpha }_{i_1}}}\right)\cdots (1+s_{i_p})\left(\frac{q-q^{-1}{X}^{-{\alpha }_{i_1}}}{1-X^{-{\alpha }_{i_1}}}\right)\text{.}$$ Thus $$U^{+}{X}^{\lambda }{U}^{+}=Q_{\lambda }{U}^{+},\text{where}{Q}_{\lambda }=\sum _{w\in W}{X}^{w\lambda }{c}_w,$$ and the $c_w$ are some linear combinations of products of terms of the form $(q-q^{-1}{X}^{\alpha })/(1-X^{\alpha })$ for roots $\alpha \in R\text{.}$ By the Satake isomorphism we know that in fact $Q_{\lambda }$ must be a symmetric function and so $$Q_{\lambda }=\sum _{w\in W}w\left(X^{\lambda }{c}_{w_0}\right),$$ where $w_0$ denotes the longest element. It is not difficult to calculate $c_{w_0}$ as it appears for a unique term in the expansion of $U^{+}$ in terms of linear combination of products of the $(q^{-1}+T_i)\text{.}$ If $w_0=s_{i_1}\cdots s_{i_p}$ is a reduced word for $w_0$ then $$\begin{array}{ccc}{w}_0{c}_{w_0}& =& \frac{q^{\ell \left(w_0\right)}}{W\left(q^2\right)}{s}_{i_1}\left(\frac{q-q^{-1}{X}^{-{\alpha }_{i_1}}}{1-X^{-{\alpha }_{i_1}}}\right)\cdots s_{i_p}\left(\frac{q-q^{-1}{X}^{-{\alpha }_{i_p}}}{1-X^{-{\alpha }_{i_p}}}\right)\\ & =& s_{i_1}\cdots s_{i_p}\left(\frac{q-q^{-1}{X}^{-s_{i_p}\cdots s_{i_2}{\alpha }_{i_1}}}{1-X^{-s_{i_p}\cdots s_{i_2}{\alpha }_{i_1}}}\right)\left(\frac{q-q^{-1}{X}^{-s_{i_p}\cdots s_{i_3}{\alpha }_{i_2}}}{1-X^{-s_{i_p}\cdots s_{i_3}{\alpha }_{i_2}}}\right)\cdots \left(\frac{q-q^{-1}{X}^{-{\alpha }_{i_p}}}{1-X^{-{\alpha }_{i_p}}}\right)\\ & =& \frac{q^{\ell \left(w_0\right)}}{W\left(q^2\right)}{w}_0\prod _{\alpha >0}\frac{q-q^{-1}{X}^{-\alpha }}{1-X^{-\alpha }},\end{array}$$ by [Bou, Prop. ????]. Thus $$Q_{\lambda }=\frac{q^{\ell \left(w_0\right)}}{W\left(q^2\right)}\sum _{w\in W}w\left(X^{\lambda }\prod _{\alpha >0}\frac{q-q^{-1}{X}^{-\alpha }}{1-X^{-\alpha }}\right)\text{.}$$ $\square$

(Straightening for the $P_{\lambda }\text{)}$ Let $\lambda \in P$ such that $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0\text{.}$ Then $$P_{s_i\lambda }=\text{????.}$$

		Proof.
	$\square$

The bar involution on the affine Hecke algebra is the $ℂ\text{-linear}$ involution $\stackrel{‾}{}:\stackrel{\sim }{H}\to \stackrel{\sim }{H}$ given by $$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_{w^{-1}}^{-1},\text{for}\hspace{0.17em}w\in \stackrel{\sim }{W}\text{.}$$ Define $C_w^{-}\in \stackrel{\sim }{H}$ by $$\overline{C_w^{-}}=C_w^{-}\text{and}{C}_w^{-}=T_w\hspace{0.17em}\text{mod}\hspace{0.17em}{q}^{-1}{L}^{-},\text{where}{L}^{-}=\mathbb{Z}{\left[q\right]}^{-1}\text{-span}\left\{T_w\hspace{0.17em}\right|\hspace{0.17em}w\in \stackrel{\sim }{W}\}\text{.}$$

Let $w_0$ be the longest element of $W\text{.}$ Let $\lambda \in P$ and let $z\in {ℂ\left[P\right]}^W\text{.}$ $$\overline{X^{\lambda }}=T_{w_0}{X}^{w_0\lambda }{T}_{w_0}^{-1},\overline{1_0}=1_0,\overline{{\epsilon }_0}={\epsilon }_0,\overline{A_{\lambda +\rho }}=A_{\lambda +\rho }\text{and}\bar{z}=z\text{.}$$

		Proof.
	The length function on $\stackrel{\sim }{W}$ can be given by $$\ell \left(w{t}_{\lambda }\right)=\sum _{\alpha >0}\mid ⟨\lambda ,{\alpha }^{\vee }⟩+\chi \left(w\alpha \right)\mid ,\text{for}\hspace{0.17em}w\in W\hspace{0.17em}\text{and}\hspace{0.17em}\lambda \in P\text{.}$$ Thus, if $\lambda \in P^{+},$ then $$T_{w_0}{T}_{t_{\lambda }}=T_{w_0{t}_{\lambda }}=T_{t_{w_0\lambda }{w}_0}=T_{t_{w_0\lambda }}{T}_{w_0}\text{.}$$ Let $\lambda \in P$ and write $\lambda =\mu -\nu$ with $\mu ,\nu \in P^{+}\text{.}$ Then, since $-w_0\mu \in P^{+}$ and $-w_0\nu \in P^{+},$ $$\overline{X^{\lambda }}=\overline{T_{t_{\mu }}{T}_{t_{\nu }}^{-1}}=T_{t_{-\mu }}^{-1}{T}_{t_{-\nu }}=T_{w_0}{T}_{t_{-w_0\mu }}^{-1}{T}_{t_{-w_0\nu }}{T}_{w_0}^{-1}=T_{w_0}{\left(X^{-w_0\lambda }\right)}^{-1}{T}_{w_0}^{-1}=T_{w_0}{X}^{w_0\lambda }{T}_{w_0}^{-1},$$ (b) Since $T_{s_i}\overline{U^{+}}=\overline{T_{s_i}^{-1}{U}^{+}}=\overline{q^{-1}{U}^{+}}=q\overline{U^{+}}$ and ${\left(\overline{U^{+}}\right)}^2=\overline{{\left(U^{+}\right)}^2}=\overline{U^{+}},$ it follows that $\overline{U^{+}}=U^{+}\text{.}$ The proof for $U^{-}$ is similar. (c) Since $z\in {ℂ\left[P\right]}^W$ is $W\text{-symmetric}$ and the coefficients are bar invariant, $z$ is an element of $Z\left(\stackrel{\sim }{H}\right)$ and part (a) implies that $\bar{z}=T_{w_0}z{T}_{w_0}^{-1}=z\text{.}$ (d) By (a), (b) and ???, $\overline{A_{\lambda +\rho }}=\overline{U^{-}{X}^{\lambda +\rho }{U}^{+}}=U^{-}{T}_{w_0}{X}^{w_0(\lambda +\rho )}{T}_{w_0}^{-1}{U}^{+}={(-q^{-1})}^{\ell \left(w_0\right)}{U}^{-}{X}^{w_0(\lambda +\rho )}{U}^{+}{q}^{\ell \left(w_0\right)}={(-1)}^{\ell \left(w_0\right)}{A}_{w_0(\lambda +\rho )}=A_{\lambda +\rho }\text{.}$ $\square$

If $\lambda \in P^{+}$ let $n_{\lambda }$ be the element of $\stackrel{\sim }{W}$ of maximal length in the double coset $W{t}_{\lambda }W\text{.}$ Then $$C_{n_{\lambda }}^{-}=q^{-\ell \left(w_0\right)}W\left(q^2\right)s_{\lambda }{1}_0\text{.}$$

		Proof.
	The first equality follows from ??? and the identity $\overline{q^{-\ell \left(w_0\right)}W\left(q^2\right)}=q^{-\ell \left(w_0\right)}W\left(q^2\right)\text{.}$ Let $\lambda \in P^{+}\text{.}$ Let $W_{\lambda }=\text{Stab}\left(\lambda \right)$ and let $w_0$ and $w_{0,\lambda }$ be the maximal length elements in $W$ and $W_{\lambda }$ respectively. Let $m_{\lambda }$ (resp. $n_{\lambda }\text{)}$ be the minimal (resp. maximal) length element in the double coset $W{t}_{\lambda }W\text{.}$ $$\begin{array}{ccc}{P}_{\lambda }(X,q^{-2})1_0& =& \frac{W\left(q^{-2}\right)}{W_{\lambda }\left(q^{-2}\right)}{1}_0{X}^{\lambda }{1}_0=\frac{W\left(q^{-2}\right)}{W_{\lambda }\left(q^{-2}\right)}\frac{1}{W{\left(q^2\right)}^2}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)T_{t_{\lambda }}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)\\ & =& \frac{q^{-2\ell \left(w_0\right)+2\ell \left(w_{\lambda }\right)}}{W_{\lambda }\left(q^2\right)W\left(q^2\right)}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)T_{m_{\lambda }{w}_0{w}_{\lambda }}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)\\ & =& \frac{q^{-\ell \left(w_0\right)+\ell \left(w_{\lambda }\right)-\ell \left(m_{\lambda }\right)}}{W_{\lambda }\left(q^2\right)W\left(q^2\right)}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)q^{\ell \left(m_{\lambda }\right)}{T}_{m_{\lambda }}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)\\ & =& \frac{q^{-\ell \left(w_0\right)+\ell \left(w_{\lambda }\right)-\ell \left(m_{\lambda }\right)}}{W\left(q^2\right)}\left(\sum _{w\in W_{\lambda }}{q}^{\ell \left(w\right)}{T}_w\right)q^{\ell \left(m_{\lambda }\right)}{T}_{m_{\lambda }}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w\right)\\ & =& \frac{q^{-\ell \left(w_0\right)+\ell \left(w_{\lambda }\right)-\ell \left(m_{\lambda }\right)+\ell \left(n_{\lambda }\right)}}{W\left(q^2\right)}\left(\sum _{x\in W{t}_{\lambda }W}{q}^{\ell \left(x\right)-\ell \left(n_{\lambda }\right)}{T}_x\right)\\ & =& \frac{q^{\ell \left(w_0\right)}}{W\left(q^2\right)}\left(\sum _{x\in W{t}_{\lambda }W}{q}^{\ell \left(x\right)-\ell \left(n_{\lambda }\right)}{T}_x\right)\end{array}$$ where the powers of $q$ inside the sum are all nonpositive. Thus $$q^{-\ell \left(w_0\right)}W\left(q^2\right)s_{\lambda }{U}^{+}=\sum _{\mu \le \lambda }{K}_{\lambda \mu }\left(q^{-2}\right)\sum _{x\in W{t}_{\mu }W}{q}^{\ell \left(x\right)-\ell \left(n_{\mu }\right)}{T}_x\text{.}$$ By ???, $K_{\lambda \mu }\left(q^{-2}\right)$ is a polynomial in $q^{-2}$ and $K_{\lambda \lambda }\left(q^{-2}\right)=1\text{.}$ $\square$

For all $z,z\prime \in W,$ $$K_{\lambda \mu }\left(q\right)=q^{⟨\lambda -\mu ,{\rho }^{\vee }⟩}{P}_{z{n}_{\mu }z\prime ,n_{\lambda }}\left(q^{-1}\right),$$ where $P_{vw}\left(q\right)$ denotes the Kazhdan-Lusztig polynomial for the affine Weyl group $\stackrel{\sim }{W}\text{.}$

Fock space from Hecke algebras

If $\nu \in P^{+}$ let $W_{\nu }$ be the stabilizer of $\nu$ and let $\stackrel{\sim }{W}/W_{\nu }$ be the set of minimal length coset representatives of cosets of $W_{\nu }\text{.}$ Define $$1_{\nu }=\frac{1}{W_{\nu }\left(q^2\right)}\sum _{w\in W_{\nu }}{T}_w,$$ and, for each $\mu \in \stackrel{\sim }{W}\nu$ let $$T_{\mu }=T_w{1}_{\nu },\text{where}\hspace{0.17em}\mu =w\nu \hspace{0.17em}\text{with}\hspace{0.17em}w\in \stackrel{\sim }{W}/W_{\nu }\text{.}$$ The elements $T_{\mu },$ $\mu \in {\stackrel{\sim }{W}}_{\nu },$ form a basis of the $\stackrel{\sim }{H}\text{-module}$ $\stackrel{\sim }{H}{1}_{\mu }\text{.}$ Define a $\mathbb{Z}\left[q\right]\text{-lattice}$ in $\stackrel{\sim }{H}{1}_{\nu }$ by $$L_{\nu }=\mathbb{Z}\left[q\right]\text{-span}\left\{T_{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {\stackrel{\sim }{W}}_{\nu }\}$$ and define $C_{\mu }^{-},$ $\mu \in {\stackrel{\sim }{W}}_{\nu },$ by $$\overline{C_{\mu }^{-}}=C_{\mu }^{-}\text{and}{C}_{\mu }^{-}=\mu \hspace{0.17em}\text{mod}\hspace{0.17em}q{L}_{\nu }\text{.}$$ The parabolic Kazhdan-Lusztig polynomials are given by $$C_{\mu }^{-}=\sum _{\lambda \in {\stackrel{\sim }{W}}_{\nu }}{P}_{\lambda \mu }^{-}(-q)T_{\lambda }\text{.}$$

$$P_{\lambda \mu }^{-}=\sum _{z\in W_{\nu }}{(-q^{-1})}^{\ell \left(z\right)}{P}_{uz,v},$$ where $\mu =u\nu$ and $\lambda =v\nu ,$ $u,v\in \stackrel{\sim }{W}/W_{\nu }\text{.}$

		Proof.
	$\square$

Fix $\ell \in {\mathbb{Z}}_{>0}$ and define the $\ell \text{-alcove}$ $A_{\ell }$ by $$A_{\ell }=\{\nu \in P^{+}\hspace{0.17em}|\hspace{0.17em}⟨\nu ,{\alpha }_i^{\vee }⟩\ge 0,\hspace{0.17em}1\le i\le n,\hspace{0.17em}\text{and}\hspace{0.17em}⟨\nu ,{\alpha }_0^{\vee }⟩<\ell \}\text{.}$$ Let $${\epsilon }_0=\frac{1}{W_0\left(q^2\right)}\sum _{w\in W}{(-q^{-1})}^{\ell \left(w\right)}{T}_w\text{so that}{\epsilon }_0^2={\epsilon }_0\text{and}{T}_i{\epsilon }_0={(-q)}^{-1}{\epsilon }_0\text{.}$$ Define $$ℱ=\underset{\nu \in A_{\ell }}{⨁}{\epsilon }_0\stackrel{\sim }{H}{1}_{\nu }\text{and}ℒ=\mathbb{Z}\left[q\right]\text{-span}\left\{|\lambda +\rho ⟩\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\},$$ where, for $\lambda \in P,$ $$|\lambda ⟩={\epsilon }_0{X}^{\beta }{T}_w{1}_{\nu },\text{if}\lambda =\ell \beta +w\nu \text{with}\hspace{0.17em}\nu \in A_{\ell },\hspace{0.17em}\beta \in P,\hspace{0.17em}w\in \stackrel{\sim }{W}/W_{\nu }\text{.}$$

(Straightening for $|\lambda ⟩\text{)}$ Assume $⟨\mu +w\nu ,{\alpha }_i^{\vee }⟩\ge 0\text{.}$ Then

(a)	If $s_{i}w\nu =w\nu$ then $(1+q)|\ell \mu +w\nu ⟩=-(1+q)|\ell s_i\mu +s_{i}w\nu ⟩,$
(b)	If $\mu =s_i\mu$ then $2|\ell \mu +w\nu ⟩=-2q|\ell s_i\mu +s_{i}w\nu ⟩,$
(c)	In general $|\ell \mu +w\nu ⟩=-q|\ell s_i\mu +s_{i}w\nu ⟩-|\ell s_i\mu +w\nu ⟩-q|\ell \mu +s_{i}w\nu ⟩\text{.}$

		Proof.
	If $⟨\mu +w\nu ,{\alpha }_i^{\vee }⟩\ge 0$ then $$\begin{array}{ccc}0& =& {\epsilon }_0(q^{-1}+T_i)(X^{\mu }+X^{s_i\mu })T_w{1}_{\nu }\\ & =& {\epsilon }_0(X^{\mu }+X^{s_i\mu })(q^{-1}+T_i)1_{\nu }\\ & =& q^{-1}|\ell \mu +w\nu ⟩+|s_i\ell \mu +s_{i}w\nu ⟩+q^{-1}|\ell s_i\mu +w\nu ⟩+|\ell \mu +s_{i}w\nu ⟩\text{.}\end{array}$$ $\square$

$\left\{|\lambda +\rho ⟩\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ is a basis of $ℱ\text{.}$

		Proof.
	$\square$

Using ??? $$\overline{|\lambda ⟩}=\overline{{\epsilon }_0{X}^{\mu }{T}_w{1}_{\nu }}={\epsilon }_0{T}_{w_0}^{-1}{X}^{w_0\lambda }{T}_{w_0}{T}_{w^{-1}}^{-1}{1}_{\nu }={\epsilon }_0{T}_{w_0}^{-1}{X}^{w_0\lambda }{T}_{w_{0}w}{1}_{\nu }={(-1)}^{\ell \left(w_0\right)}{q}^{-\ell \left(w_0\right)-\ell \left(w_{\nu }\right)}|w_0\lambda ⟩$$ For $\lambda \in P^{+}$ define $G_{\lambda +\rho }^{-}$ by $$\overline{G_{\lambda +\rho }^{-}}=G_{\lambda +\rho }^{-}\text{and}{G}_{\lambda +\rho }^{-}=|\lambda +\rho ⟩\hspace{0.17em}\text{mod}\hspace{0.17em}qℒ\text{.}$$

$$G_{\lambda +\rho }^{-}=\sum _{\mu \in P^{+}}{P}_{\lambda \mu }^{-}(-q)|\mu +\rho ⟩,$$ where $P_{\lambda \mu }^{-}$ is the parabolic Kazhdan-Lusztig poylnomial.

		Proof.
	$\square$

(Steinberg’s tensor product theorem) Let $\lambda \in P^{+}\text{.}$ Then $$G_{\lambda +\rho }^{-}=S_{{\lambda }^{\left(1\right)}}{G}_{{\lambda }^{\left(0\right)}+\rho }^{-},\text{where}\lambda ={\lambda }^{\left(0\right)}+\ell {\lambda }^{\left(1\right)},$$ with ${\lambda }^{\left(0\right)}$ $n\text{-restricted.}$

		Proof.
	$\square$

Let $U_{\xi }𝔤$ be the quantum group at the value $\xi \text{.}$ For each $\lambda \in P^{+}$ let $\Delta \left(\lambda \right)$ be the Weyl module of highest weight $\lambda$ and let $L\left(\lambda \right)$ be the simple module of highest weight $\lambda \text{.}$

(Lusztig conjecture) Let $\xi$ be a primitive ${\ell }^{\text{th}}$ root of unity. Define a vector space isomorphism $$\begin{array}{cccc}\Phi :& R\left(U_{\xi }𝔤\right)& ⟶& {ℱ}_{\ell }\\ & \left[\Delta \left(\lambda \right)\right]& ⟼& |\lambda +\rho ⟩\text{.}\end{array}\text{Then}\Phi \left(\left[L\left(\lambda \right)\right]\right)=G_{\lambda +\rho }^{-}\text{.}$$

		Proof.
	$\square$

Crystals

Let $C={\left(⟨{\alpha }_i,{\alpha }_j^{\vee }⟩\right)}_{i\in I}$ be a Cartan matrix. Define free abelian groups $$P=\sum _{i\in I}\mathbb{Z}{\omega }_i\text{and}Q=\sum _{i\in I}\mathbb{Z}{\alpha }_i,\text{and a pairing}⟨,⟩:P\times Q\to \mathbb{Z}$$ given by $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}\text{.}$

A crystal is a set $B$ with maps $$\begin{array}{cc}& \text{wt}:B\to P\\ {\epsilon }_i:B\to \mathbb{Z}\cup \{-\infty \}& \text{and}& {\phi }_i:B\to \mathbb{Z}\cup \{-\infty \},\\ {\stackrel{\sim }{e}}_i:B\to B\cup \left\{0\right\}& \text{and}& {\stackrel{\sim }{f}}_i:B\to B\cup \left\{0\right\},\end{array}$$ such that

(1)	If ${\stackrel{\sim }{e}}_{i}b\ne 0$ then $$\text{wt}\left({\stackrel{\sim }{e}}_{i}b\right)=\text{wt}\left(b\right)+{\alpha }_i,{\epsilon }_i\left({\stackrel{\sim }{e}}_{i}b\right)={\epsilon }_i\left(b\right)-1,{\phi }_i\left({\stackrel{\sim }{e}}_{i}b\right)={\phi }_i\left(b\right)+1,{\stackrel{\sim }{f}}_i{\stackrel{\sim }{e}}_{i}b=b,$$ and if ${\stackrel{\sim }{f}}_{i}b\ne 0$ the $$\text{wt}\left({\stackrel{\sim }{f}}_{i}b\right)=\text{wt}\left(b\right)-{\alpha }_i,{\epsilon }_i\left({\stackrel{\sim }{f}}_{i}b\right)={\epsilon }_i\left(b\right)+1,{\phi }_i\left({\stackrel{\sim }{f}}_{i}b\right)={\phi }_i\left(b\right)-1,{\stackrel{\sim }{e}}_i{\stackrel{\sim }{f}}_{i}b=b,$$
(2)	${\phi }_i\left(b\right)={\epsilon }_i\left(b\right)+⟨\text{wt}\left(b\right),{\alpha }_i^{\vee }⟩,$ and
(3)	If ${\phi }_i\left(b\right)=-\infty$ then ${\stackrel{\sim }{e}}_{i}b={\stackrel{\sim }{f}}_{i}b=0\text{.}$

The crystal graph of $B$ is the graph with $$\text{vertex set}\hspace{0.17em}B\text{and}\text{labeled edges}b\stackrel{i}{⟷}{\stackrel{\sim }{e}}_{i}b\text{when}\hspace{0.17em}{\stackrel{\sim }{e}}_{i}b\ne 0\text{.}$$ The $\mu \text{-weight}$ space of a crystal $B$ is the set $$B_{\mu }=\{b\in B\hspace{0.17em}|\hspace{0.17em}\text{wt}\left(b\right)=\mu \}\text{.}$$ The character of $B$ is the weight generating function of $B,$ $${\chi }^B=\sum _{b\in B}{e}^{\text{wt}\left(b\right)}=\sum _{\mu \in P}\text{Card}\left(B_{\mu }\right)e^{\mu }\in \mathbb{Z}\left[P\right]\text{.}$$ A normal crystal is a crystal $B$ such that $${\epsilon }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{e}}_i^{k}b\ne 0\}\text{and}{\phi }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{f}}_i^{k}b\ne 0\}\text{.}$$ If $B$ is a normal crystal and $b\in B$ the $i\text{-string}$ of $b$ is the set $$\left\{{\stackrel{\sim }{f}}_i^{{\phi }_i\left(b\right)}b\stackrel{i}{⟷}\cdots \stackrel{i}{⟷}{\stackrel{\sim }{f}}_i^{2}b\stackrel{i}{⟷}{\stackrel{\sim }{f}}_{i}b\stackrel{i}{⟷}b\stackrel{i}{⟷}{\stackrel{\sim }{e}}_{i}b\stackrel{i}{⟷}{\stackrel{\sim }{e}}_i^{2}b\stackrel{i}{⟷}\cdots \stackrel{i}{⟷}{\stackrel{\sim }{e}}_i^{{\epsilon }_i\left(b\right)}b\right\},$$ and (3) is equivalent to $⟨\text{wt}\left({\stackrel{\sim }{e}}_i^{{\epsilon }_i\left(b\right)}b\right),{\alpha }_i^{\vee }⟩=-⟨\text{wt}\left({\stackrel{\sim }{f}}_i^{\phi \left(b\right)}b\right),{\alpha }_i^{\vee }⟩$ so that every $i$ string in a normal crystal $B$ is a model for a finite dimensional ${𝔰𝔩}_2\text{-module.}$

If $B$ is a normal crystal define a bijection $s_i:B\to B$ by $$s_{i}b=\{\begin{array}{cc}{\stackrel{\sim }{f}}_i^{{\text{wt}}_i\left(b\right)}b,& \text{if}\hspace{0.17em}{\text{wt}}_i\left(b\right)\ge 0,\\ {\stackrel{\sim }{e}}_i^{-{\text{wt}}_i\left(b\right)}b,& \text{if}\hspace{0.17em}{\text{wt}}_i\left(b\right)\le 0,\end{array}\text{so that}\text{wt}\left(s_{i}b\right)=s_i\text{wt}\left(b\right),\text{for all}\hspace{0.17em}b\in B\text{.}$$ The map $s_i$ flips each $i\text{-string}$ in $B\text{.}$ The equality $\text{wt}\left(s_{i}b\right)=s_i\text{wt}\left(b\right)$ implies $${\chi }^B\in \mathbb{Z}{\left[P\right]}^W,\text{for any normal crystal}\hspace{0.17em}B\text{.}$$

[Kashiwara, Duke 73 (1994), 383-413] Let $B$ be a normal crystal. The maps $s_i:B\to B$ $i\in I,$ define an action of $W$ on $B\text{.}$

		Proof.
	$\square$

Let $B_1$ and $B_2$ be crystals. A morphism $\psi \in \text{Hom}(B_1,B_2)$ is a map $\varphi :B_1\to B_2\cup \left\{0\right\}$ such that $$\text{wt}\left(\psi \left(b\right)\right)=\text{wt}\left(b\right),{\epsilon }_i\left(\psi \left(b\right)\right)={\epsilon }_i\left(b\right),{\phi }_i\left(\psi \left(b\right)\right)={\phi }_i\left(b\right),$$ and $$\text{if}\hspace{0.17em}b\stackrel{i}{⟷}{\stackrel{\sim }{e}}_{i}b\hspace{0.17em}\text{and}\hspace{0.17em}\psi \left(b\right)\ne 0\hspace{0.17em}\text{and}\hspace{0.17em}\psi \left({\stackrel{\sim }{e}}_{i}b\right)\ne 0\text{then}\psi \left(b\right)\stackrel{i}{⟷}{\stackrel{\sim }{e}}_i\psi \left(b\right)$$ A strict morphism is a morphism that commutes with all ${\stackrel{\sim }{e}}_i$ and all ${\stackrel{\sim }{f}}_i\text{.}$

The tensor product of $B_1$ and $B_2$ is the crystal $$\begin{array}{c}{B}_1\otimes B_2=\{b_1\otimes b_2\hspace{0.17em}|\hspace{0.17em}{b}_1\in B_1,b_2\in B_2\}\text{with}\\ {\text{wt}}_i(b_1\otimes b_2)={\text{wt}}_i\left(b_1\right)+{\text{wt}}_i\left(b_2\right),\begin{array}{ccc}{\epsilon }_i(b_1\otimes b_2)& =& \text{max}\{{\epsilon }_i\left(b_1\right),{\epsilon }_i\left(b_2\right)-⟨\text{wt}\left(b_1\right),{\alpha }_i^{\vee }⟩\},\\ {\phi }_i(b_1\otimes b_2)& =& \text{max}\{{\phi }_i\left(b_1\right)+⟨\text{wt}\left(b_2\right),{\alpha }_i^{\vee }⟩,{\phi }_i\left(b_2\right)\},\end{array}\\ {\stackrel{\sim }{e}}_i(b_1\otimes b_2)=\{\begin{array}{cc}{\stackrel{\sim }{e}}_i{b}_1\otimes b_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)\ge {\epsilon }_i\left(b_2\right),\\ b_1\otimes {\stackrel{\sim }{e}}_i{b}_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)<{\epsilon }_i\left(b_2\right),\end{array}\text{and}{\stackrel{\sim }{f}}_i(b_1\otimes b_2)=\{\begin{array}{cc}{\stackrel{\sim }{f}}_i{b}_1\otimes b_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)>{\epsilon }_i\left(b_2\right),\\ b_1\otimes {\stackrel{\sim }{f}}_i{b}_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)\le {\epsilon }_i\left(b_2\right),\end{array}\end{array}$$ If $B_1,B_2,B_3$ are crystals, then the map $$\begin{array}{ccc}(B_1\otimes B_2)\otimes B_3& \stackrel{\sim }{⟶}& B_1\otimes (B_2\otimes B_3)\\ (b_1\otimes b_2)\otimes b_3& ⟼& b_1\otimes (b_2\otimes b_3)\end{array}$$ is a crystal isomorphism and so we may simply write $B_1\otimes B_2\otimes B_3$ for the tensor product of $B_1,$ $B_2$ and $B_3\text{.}$

If $B_1$ and $B_2$ are normal crystals then $B_1\otimes B_2$ is normal.

		Proof.
	$\square$

If $B$ is a crystal the dual crystal is the crystal $B^{*}\left\{b^{*}\hspace{0.17em}\right|\hspace{0.17em}b\in B\}$ with $\text{wt}\left(b^{*}\right)=-\text{wt}\left(b\right),$ $$\begin{array}{ccc}{\epsilon }_i\left(b^{*}\right)={\varphi }_i\left(b\right),& \text{and}& \phi \left(b^{*}\right)={\epsilon }_i\left(b\right),\\ {\stackrel{\sim }{e}}_i\left(b^{*}\right)={\left({\stackrel{\sim }{f}}_{i}b\right)}^{*},& \text{and}& {\stackrel{\sim }{f}}_i\left(b^{*}\right)={\left({\stackrel{\sim }{e}}_{i}b\right)}^{*}\text{.}\end{array}$$ The crystal graph of $B^{*}$ is obtained by reversing all the arrows in the crystal graph of $B\text{.}$

Irreducible crystals $B\left(\lambda \right)$

A normal crystal $B$ is irreducible if the crystal graph of $B$ has a single connected component??? A highest weight path is an element $b\in B$ such that ${\stackrel{\sim }{e}}_{i}b=0$ for all $i\in I\text{.}$

The irreducible highest weight crystals $B\left(\lambda \right)$ are indexed by $\lambda \in P^{+}\text{.}$

		Proof.
	$\square$

We would like to show that there is a unique normal crystal $B\left(\lambda \right)$ of highest weight $\lambda \text{.}$ Define $$B(\lambda +\mu )\hspace{0.17em}\text{is the connected component of}\hspace{0.17em}{b}_{\lambda }^{}\otimes b_{\mu }^{+}\hspace{0.17em}\text{in}\hspace{0.17em}B\left(\lambda \right)\otimes B\left(\mu \right)\text{.}$$ Thus, by definition there is a canonical injection $$\begin{array}{cccc}{\iota }_{\lambda +\mu }^{\lambda \otimes \mu }:& B(\lambda +\mu )& \hookrightarrow & B\left(\lambda \right)\otimes B\left(\mu \right)\\ & b_{\lambda +\mu }^{+}& ⟼& b_{\lambda }^{+}\otimes b_{\mu }^{+}\end{array}$$

Let $\lambda +\mu \in P^{+}\text{.}$ The crystal $B(\lambda +\mu )$ is well defined, i.e $$B(\lambda +\mu )\cong B(\gamma +\delta )\text{if}\lambda +\mu =\gamma +\delta \text{.}$$

		Proof.
	$\square$

This reduces the problem of finding $B\left(\lambda \right)$ to the fundamental weights.

Let $\lambda \in P^{+}\text{.}$ Then $B\left(\lambda \right)$ exists.

Another characterization? $$B\left(\lambda \right)=\{b\otimes b_{\lambda }^{-\infty }\in B\left(\infty \right)\otimes B_{\lambda }^{-\infty }\hspace{0.17em}|\hspace{0.17em}{\epsilon }_i^{*}\left(b\right)\le ⟨\lambda ,{\alpha }_i^{\vee }⟩,\hspace{0.17em}\text{for all}\hspace{0.17em}i\in I\}\text{.}$$

For each $m\in {\mathbb{Z}}_{>0}$ and each $\lambda \in P^{+}$ there unique injective maps $$\begin{array}{cccc}{S}_m:& B\left(\lambda \right)& ⟶& B\left(m\lambda \right)\\ & b_{\lambda }^{+}& ⟼& b_{m\lambda }^{+}\end{array}$$ such that $$\begin{array}{c}\text{wt}\left(S_{m}b\right)=m\text{wt}\left(b\right),{\epsilon }_i\left(S_{m}b\right)=m{\epsilon }_i\left(b\right),\phi \left(S_{m}b\right)=m{\phi }_i\left(m\right),\text{and}\\ S_m\left({\stackrel{\sim }{e}}_{i}b\right)={\stackrel{\sim }{e}}_i^m{S}_m\left(b\right)\text{and}{S}_m\left({\stackrel{\sim }{f}}_{i}b\right)={\stackrel{\sim }{f}}_i^m{S}_m\left(b\right)b\text{.}\end{array}$$

$${B\left(\lambda \right)}^{\vee }=B(-w_0\lambda )\text{.}$$ The crystal $B\left(\infty \right)$ Define projections ${\pi }_{\lambda }^{\lambda +\mu }:B(\lambda +\mu )\to B\left(\lambda \right)$ by the composition $$\begin{array}{cccccc}{\pi }_{\lambda }^{\lambda +\mu }:& B(\lambda +\mu )& \hookrightarrow & B\left(\lambda \right)\otimes B\left(\mu \right)& ⟶& B\left(\lambda \right)\\ & & & b\otimes b_{\mu }^{+}& ⟼& b,\\ & & & b\otimes b\prime & ⟼& 0,& \text{if}\hspace{0.17em}b\prime \ne b_{\mu }^{+}\text{.}\end{array}$$ The projective system defined by the ${\pi }_{\lambda }^{\lambda +\mu }$ allows us to define $$B\left(\infty \right)=\underset{⟵}{\text{lim}}B\left(\lambda \right)\text{so that}{\pi }_{\lambda }:B\left(\infty \right)\to B\left(\lambda \right),\text{is such that}{\pi }_{\lambda }^{\lambda +\mu }{\pi }_{\lambda +\mu }={\pi }_{\lambda },$$ for all $\lambda \in P^{+}\text{.}$ For each $j\in I$ define a crystal $$B_j\left(\mathbb{Z}\right)=\left\{b_j\left(n\right)\hspace{0.17em}\right|\hspace{0.17em}n\in \mathbb{Z}\}\cdots \stackrel{j}{⟷}{b}_j(-1)\stackrel{j}{⟷}{b}_j\left(0\right)\stackrel{j}{⟷}{b}_j\left(1\right)\stackrel{j}{⟷}\cdots$$ with $$\text{wt}\left(b_j\left(n\right)\right)=n{\alpha }_j{\epsilon }_i\left(b_j\left(n\right)\right)=\{\begin{array}{cc}-n,& \text{if}\hspace{0.17em}i=j,\\ -\infty ,& \text{if}\hspace{0.17em}i\ne j,\end{array}{\phi }_i\left(b_j\left(n\right)\right)=\{\begin{array}{cc}n,& \text{if}\hspace{0.17em}i=j,\\ -\infty ,& \text{if}\hspace{0.17em}i\ne j,\end{array}$$ and $${\stackrel{\sim }{e}}_i\left(b_j\left(n\right)\right)=\{\begin{array}{cc}{b}_j(n+1),& \text{if}\hspace{0.17em}i=j,\\ 0,& \text{if}\hspace{0.17em}i\ne j,\end{array}\text{and}{\stackrel{\sim }{f}}_i\left(b_j\left(n\right)\right)=\{\begin{array}{cc}{b}_j(n-1),& \text{if}\hspace{0.17em}i=j,\\ 0,& \text{if}\hspace{0.17em}i\ne j\text{.}\end{array}$$

(a)

For each $j\in I$ there is a crystal injection

(b)

Let $(i_1,i_2,\dots )$ be a sequence of elements of $I$ such that each $i\in I$ appears an infinite number of times. Then the subcrystal of $\cdots \otimes B_{i_2}\left(\mathbb{Z}\right)\otimes B_{i_1}\left(\mathbb{Z}\right)$ given by $$B\left(\infty \right)=\{\cdots \otimes b_{i_2}(-a_2)\otimes b_{i_1}(-a_1)\hspace{0.17em}|\hspace{0.17em}{a}_i\in {\mathbb{Z}}_{\ge 0},\hspace{0.17em}{a}_k=0\hspace{0.17em}\text{for}\hspace{0.17em}k\gg 0\},$$ the subcrystal of $\cdots \otimes B_{i_2}\left(\mathbb{Z}\right)\otimes B_{i_1}\left(\mathbb{Z}\right)$ generated by $\cdots b_{i_2}\left(0\right)\otimes b_{i_1}\left(0\right)\text{.}$

If $\{i_1,i_2,\dots \}$ is a sequence of elements of $I$ such that each $i\in I$ appears an infinite number of times the composition $\cdots {\Phi }_{i_3}\circ {\Phi }_{i_2}\circ {\Phi }_{i_1}$ realize $$B\left(\infty \right)=\{\cdots \otimes b_{i_2}(-a_2)\otimes b_{i_1}(-a_1)\hspace{0.17em}|\hspace{0.17em}{a}_i\in {\mathbb{Z}}_{\ge 0},\hspace{0.17em}{a}_k=0\hspace{0.17em}\text{for}\hspace{0.17em}k\gg 0\}\text{.}$$

For each $\lambda \in P^{+},$ define a crystal $B_{\lambda }^{-\infty }=\left\{b_{\lambda }^{-\infty }\right\}$ with $$\text{wt}\left(b_{\lambda }^{-\infty }\right)=\lambda ,{\epsilon }_i\left(b_{\lambda }^{-\infty }\right)=-\infty ,{\phi }_i\left(b_{\lambda }^{-\infty }\right)=-\infty ,{\stackrel{\sim }{e}}_i\left(b_{\lambda }^{-\infty }\right)=0,{\stackrel{\sim }{f}}_i\left(b_{\lambda }^{-\infty }\right)=0,$$ for all $i\in I\text{.}$ Then $$B\left(\lambda \right)\hspace{0.17em}\text{is the normal subcrystal of}\hspace{0.17em}B\left(\infty \right)\otimes B_{\lambda }^{-\infty }\hspace{0.17em}\text{generated by}\hspace{0.17em}{b}_0^{+}\otimes b_{\lambda }^{-\infty }\text{.}$$

Define $*:B\left(\infty \right)\to B\left(\infty \right)$ to be the unique involution such that $${\Phi }_i\left({\left({\stackrel{\sim }{f}}_i^n{b}^{*}\right)}^{*}\right)=b\otimes {\stackrel{\sim }{f}}_i^n{b}_i\left(0\right),\text{for all}\hspace{0.17em}i\in I\text{.}$$ THIS DEFINITION NEEDS REWORKING!

Representation crystals

The quantum group is the $\mathbb{Q}\left(q\right)$ algebra given by generators $$E_i,F_i,K_i,K_i^{-1},i\in I,$$ with relations $$\begin{array}{cc}\multicolumn{2}{c}{E_i{F}_j-F_j{E}_i={\delta }_{ij}\left(\frac{K_i-K_i^{-1}}{q-q^{-1}}\right)}\\ K_i{K}_i^{-1}=K_i^{-1}{K}_i=1,& K_i{K}_j=K_j{K}_i,\\ K_i{E}_j=q^{⟨{\alpha }_j,{\alpha }_i^{\vee }⟩}{E}_j{K}_i,& K_i{F}_j=q^{-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩}{F}_j{K}_i,\\ 0=\sum _{r=0}^{{\ell }_{ij}}\left[\genfrac{}{}{0ex}{}{{\ell }_{ij}}{r}\right]E_i^r{E}_j{E}_i^{{\ell }_{ij}-r},& 0=\sum _{r=0}^{{\ell }_{ij}}\left[\genfrac{}{}{0ex}{}{{\ell }_{ij}}{r}\right]F_i^r{F}_j{F}_i^{{\ell }_{ij}-r},\text{for}\hspace{0.17em}i\ne j,\end{array}$$ where ${\ell }_{ij}=-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩+1,$ and $$\left[k\right]=\frac{q^k-q^{-k}}{q-q^{-1}},\left[k\right]!=\left[k\right][k-1]\cdots \left[2\right]\left[1\right],\left[\genfrac{}{}{0ex}{}{n}{k}\right]=\frac{\left[n\right]!}{\left[k\right]![n-k]!}\text{.}$$

(Drinfel’d) The algebra $U_q𝔤$ is the unique Cartan preserving Hopf algebra deformation of $G\text{.}$

		Proof.
	$\square$

An integrable $U_q𝔤\text{-module}$ is a $U_q𝔤\text{-module}$ $M$ such that $$M=\underset{\mu \in P}{⨁}{M}_{\mu },\text{where}{M}_{\mu }=\{m\in M\hspace{0.17em}|\hspace{0.17em}{K}_{i}m=q^{⟨\mu ,{\alpha }_i^{\vee }⟩}m,\hspace{0.17em}\text{for all}\hspace{0.17em}i\in I\},$$ and for each $m\in M$ and $i\in I,$ $E_i^{k}m=0$ and $F_i^{k}m=0$ for $k\gg 0\text{.}$

There is a bijection $$\begin{array}{ccc}\left\{\text{simple integrable}\hspace{0.17em}{U}_q𝔤\text{-modules}\right\}& \stackrel{1-1}{⟷}& P^{+}\\ L\left(\lambda \right)& \leftrightarrow & \lambda \end{array}$$

		Proof.
	$\square$

Let $M$ be a integrable $U_q𝔤\text{-module.}$ The crystal operators ${\stackrel{\sim }{e}}_i:M\to M$ and ${\stackrel{\sim }{f}}_i:M\to M$ are the linear operators determined by $${\stackrel{\sim }{e}}_i\left(F_i^{\left(k\right)}m\right)=F_i^{(k-1)}m\text{and}{\stackrel{\sim }{f}}_i\left(F_i^{\left(k\right)}m\right)=F_i^{(k+1)}m,$$ for all $k\in {\mathbb{Z}}_{\ge 0}$ and $m\in M$ such that $E_{i}m=0$ and $F_i^{\left(k\right)}m\ne 0\text{.}$ The convention is that $F_i^{(-1)}m=0\text{.}$

A crystal basis of $M$ is a pair $(L,B),$ $$L=\underset{\mu \in P}{⨁}{L}_{\mu },B=\underset{\mu \in P}{⨆}{B}_{\mu },\text{where}\begin{array}{c}{L}_{\mu }\hspace{0.17em}\text{is a free}\hspace{0.17em}\mathbb{Q}\left[q\right]\text{-module with}\\ M_{\mu }=\mathbb{Q}\left(q\right){\otimes }_{\mathbb{Q}\left[q\right]}{L}_{\mu },\\ B_{\mu }\hspace{0.17em}\text{is a basis of}\hspace{0.17em}{L}_{\mu }/q{L}_{\mu },\end{array}$$ for all $\mu \in P,$ and such that $L$ is stable under ${\stackrel{\sim }{e}}_i$ and ${\stackrel{\sim }{f}}_i$ and the images of the operators ${\stackrel{\sim }{e}}_i$ and ${\stackrel{\sim }{f}}_i$ on $L_{\mu }/q{L}_{\mu }$ with the definitions $$\begin{array}{c}\text{wt}\left(b\right)=\mu ,\text{if}\hspace{0.17em}b\in B_{\mu },\\ {\epsilon }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{e}}_i^{k}b\ne 0\},\text{and}{\phi }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{f}}_i^{k}b\ne 0\},\end{array}$$ make $B$ into a crystal.

For each $\lambda \in P^{+}$ let $L\left(\lambda \right)$ be the irreducible $U_q𝔤\text{-module}$ of highest weight $\lambda$ and fix a highest weight vector $v_{\lambda }^{+}$ in $L\left(\lambda \right)\text{.}$ Define homomorphisms of $U_q^{-}𝔤\text{-modules}$ $$\begin{array}{cccc}{\pi }_{\lambda }:& U_q^{-}𝔤& ⟶& L\left(\lambda \right)\\ & u& ⟼& u{v}_{\lambda }^{+}\end{array}$$ Define $$\begin{array}{ccc}ℒ\left(\lambda \right)& =& \mathbb{Q}\left[q\right]\text{-span}\left\{{\stackrel{\sim }{f}}_{i_k}\cdots {\stackrel{\sim }{f}}_{i_1}{v}_{\lambda }^{+}\hspace{0.17em}\right|\hspace{0.17em}{i}_1,\dots ,i_k\in I\},\text{and}\\ B\left(\lambda \right)& =& \{\text{images of}\hspace{0.17em}{\stackrel{\sim }{f}}_{i_k}\cdots {\stackrel{\sim }{f}}_{i_1}{v}_{\lambda }^{+}\hspace{0.17em}\text{in}\hspace{0.17em}ℒ\left(\lambda \right)/qℒ\left(\lambda \right)\}\text{.}\end{array}$$

(a)

Let $\lambda \in P^{+}$ and let $L\left(\lambda \right)$ be the irreducible $U_q𝔤$ module of highest weight $\lambda \text{.}$ Then $(ℒ\left(\lambda \right),B\left(\lambda \right))$ is a crystal basis of $L\left(\lambda \right)\text{.}$

(b)

There is a unique crystal basis $(ℒ\left(\infty \right),B\left(\infty \right))$ of $U_q^{-}𝔤$ such that, for all $\lambda \in P^{+},$ $${\pi }_{\lambda }\left(ℒ\left(\infty \right)\right)=ℒ\left(\lambda \right)\text{and}{\bar{\pi }}_{\lambda }\left(B\left(\infty \right)\right)=B\left(\lambda \right)\cup \left\{0\right\},$$ where ${\bar{\pi }}_{\lambda }:ℒ\left(\infty \right)/qℒ\left(\infty \right)\to ℒ\left(\lambda \right)/qℒ\left(\lambda \right)$ is the map induced by ${\pi }_{\lambda }\text{.}$

Quiver crystals

Let $(I,{\Omega }^{\pm })$ be the directed graph with vertex set $I$ and an edge $(i\to j)\in {\Omega }^{\pm }$ if $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=-1\text{.}$ Fix an orientiation of $(I,{\Omega }^{\pm }),$ i.e. a map $$\begin{array}{cccc}c:& {\Omega }^{\pm }& ⟶& {ℂ}^{*}\\ & i\to j& ⟼& c_{i\to j}\end{array}\text{such that}{c}_{i\to j}+c_{j\to i}=0\text{.}$$ Recall that $$P^{+}=\sum _{i\in I}{\mathbb{Z}}_{\ge 0}{\omega }_i\text{and}{Q}^{-}=-\sum _{i\in I}{\mathbb{Z}}_{\ge 0}{\alpha }_i\text{.}$$

Fix $$\lambda =\sum _{i\in I}{\lambda }_i{\omega }_i\in P^{+}\text{and the}\hspace{0.17em}I\text{-graded vector space}{ℂ}^{\lambda }=\underset{i\in I}{⨁}{ℂ}^{{\lambda }_i}\text{.}$$ For each $$-\nu =\sum _{i\in I}-{\nu }_i{\alpha }_i\in Q^{-}\text{fix the}\hspace{0.17em}I\text{-graded vector space}{ℂ}^{\nu }=\underset{i\in I}{⨁}{ℂ}^{{\nu }_i}\text{.}$$ There is a natural ${GL}_{\nu }={\displaystyle \prod _{i\in I}}{GL}_{{\nu }_i}\left(ℂ\right)$ action on the variety $$\begin{array}{ccc}{X\left(\lambda \right)}_{\lambda -\nu }& =& \left(\underset{(i\to j)\in {\Omega }^{\pm }}{⨁}\text{Hom}({ℂ}^{{\nu }_i},{ℂ}^{{\nu }_j})\right)\oplus \left(\underset{i\in I}{⨁}\text{Hom}({ℂ}^{{\nu }_i},{ℂ}^{{\lambda }_i})\right)\oplus \left(\underset{i\in I}{⨁}\text{Hom}({ℂ}^{{\lambda }_i},{ℂ}^{{\nu }_i})\right)\\ & =& \{x=\left(\underset{(i\to j)\in {\Omega }^{\pm }}{⨁}{x}_{i\to j}\right)\oplus \left(\underset{i\in I}{⨁}{x}_{i\to }\right)\oplus \left(\underset{i\in I}{⨁}{x}_{i\leftarrow }\right)\hspace{0.17em}|\hspace{0.17em}\begin{array}{c}{x}_{i\to j}\in \text{Hom}({ℂ}^{{\nu }_i},{ℂ}^{{\nu }_j}),\\ x_{i\to }\in \text{Hom}({ℂ}^{{\nu }_i},{ℂ}^{{\lambda }_i}),\\ x_{i\leftarrow }\in \text{Hom}({ℂ}^{{\lambda }_i},{ℂ}^{{\nu }_i})\end{array}\}\end{array}$$ Write $$x_{\leftrightarrow }=\underset{i\to j}{⨁}{x}_{i\to j},x_{\to }=\underset{i\in I}{⨁}{x}_{i\to },\text{and}{x}_{\leftarrow }=\underset{i\in I}{⨁}{x}_{i\to },\text{for}\hspace{0.17em}x\in X{\left(\lambda \right)}_{\lambda -\nu }\text{.}$$ Use the orientation to define a ${GL}_{\nu }\text{-invariant}$ symplectic form on ${X\left(\lambda \right)}_{\lambda -\nu }$ by $$\omega (x,y)=\sum _{(i\to j)\in {\Omega }^{\pm }}{c}_{i\to j}\text{Tr}\left(x_{j\to i}{y}_{i\to j}\right)+\sum _{i\in I}\text{Tr}\left(x_{i\leftarrow }{y}_{i\to }\right)-\sum _{i\in I}\text{Tr}\left(y_{i\to }{x}_{i\leftarrow }\right),\text{for}\hspace{0.17em}x,y\in {X\left(\lambda \right)}_{\lambda -\nu }\text{.}$$ The corresponding moment map $$\mu :{X\left(\lambda \right)}_{\lambda -\nu }\to {𝔤𝔩}_{\nu }\text{is given by}\mu {\left(x\right)}_i=x_{i\leftarrow }{x}_{i\to }+\sum _{(i\to j)\in {\Omega }^{\pm }}{c}_{i\to j}\left(x_{j\to i}{x}_{i\to j}\right)\text{.}$$ A point $x\in {X\left(\lambda \right)}_{\lambda -\nu }$ is stable if every $x_{\leftrightarrow }\text{-invariant}$ $I\text{-graded}$ vector space $U\subseteq \text{ker}\left(x_{\mapsto }\right)$ is $0\text{.}$ Let $$\begin{array}{ccc}{{\rm X}\left(\lambda \right)}_{\lambda -\nu }& =& {\mu }^{-1}{\left(0\right)}^{\text{st}}/{GL}_{\nu }=\left\{{GL}_{\nu }\text{-orbits of stable points in}\hspace{0.17em}{\mu }^{-1}\left(0\right)\right\},\\ {\Lambda \left(\lambda \right)}_{\lambda -\nu }& =& \{\left[x\right]\in {{\rm X}\left(\lambda \right)}_{\lambda -\nu }\hspace{0.17em}|\hspace{0.17em}{x}_{\mapsto }=0\hspace{0.17em}\text{and}\hspace{0.17em}{x}_{\leftrightarrow }\hspace{0.17em}\text{is nilpotent}\},\\ {B\left(\lambda \right)}_{\lambda -\nu }& =& \left\{\text{irreducible components of}\hspace{0.17em}{\Lambda \left(\lambda \right)}_{\lambda -\nu }\right\}\text{.}\end{array}$$ If $b\in {B\left(\lambda \right)}_{\lambda -\nu }$ define $${\epsilon }_i\left(b\right)={\epsilon }_i\left(\left[x\right]\right)={\epsilon }_i\left(x\right)=\text{dim}\left(\text{coker}(x_{i\leftarrow }\oplus \underset{j\in I}{⨁}{x}_{i\leftarrow j})\right)$$ for a generic point $\left[x\right]$ in $b\text{.}$ Let $${X\left(\lambda \right)}_{\lambda -(\nu -d{\alpha }_i)}^{\lambda -\nu }=\left\{(x,V)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{c}x\in {X\left(\lambda \right)}_{\lambda -\nu },\text{im}\left(x_{\leftarrow }\right)\subseteq V\subseteq {ℂ}^{\nu },\\ \text{dim}\left(V\right)=\nu -d{\alpha }_i,V\hspace{0.17em}\text{is}\hspace{0.17em}{x}_{\leftrightarrow }\text{-stable}\end{array}\}$$ and define $$\begin{array}{ccccc}{X\left(\lambda \right)}_{\lambda -(\nu -d{\alpha }_i)}& \stackrel{q_1}{⟵}& {X\left(\lambda \right)}_{\lambda -(\nu -d{\alpha }_i)}^{\lambda -\nu }& \stackrel{q^2}{⟶}& {X\left(\lambda \right)}_{\lambda -\nu }\\ x{|}_V& \leftarrow & (x,V)& \mapsto & x\end{array}$$ Passing to ${GL}_{\nu }\text{-orbits}$ of stable points these maps induce $${{\rm X}\left(\lambda \right)}_{\lambda -(\nu -d{\alpha }_i)}^{{\epsilon }_i=0}\stackrel{q_1}{⟵}{q}_2^{-1}\left({{\rm X}\left(\lambda \right)}_{\lambda -\nu }^{{\epsilon }_i=d}\right)\stackrel{q_2}{⟶}{{\rm X}\left(\lambda \right)}_{\lambda -\nu }^{{\epsilon }_i=d}$$ where ${{\rm X}\left(\lambda \right)}_{\lambda -\nu }^{{\epsilon }_i=d}=\{\left[x\right]\in {{\rm X}\left(\lambda \right)}_{\lambda -\nu }\hspace{0.17em}|\hspace{0.17em}{\epsilon }_i\left(x\right)=d\}\text{.}$ The result is a bijection $${\stackrel{\sim }{e}}_i^d:{B\left(\lambda \right)}_{\lambda -(\nu -d{\alpha }_i)}^{{\epsilon }_i=0}\stackrel{q_2\circ q_1^{-1}}{⟶}{B\left(\lambda \right)}_{\lambda -\nu }^{{\epsilon }_i=d}\text{.}$$

Let $\lambda \in P^{+}\text{.}$ The set $$B\left(\lambda \right)=\underset{\nu \in Q^{-}}{⨆}{B\left(\lambda \right)}_{\lambda -\nu }\text{with the maps}{\stackrel{\sim }{e}}_i:B\left(\lambda \right)\to B\left(\lambda \right)\cup \left\{0\right\}$$ defined by ??? is a realization of the highest weight crystal of weight $\lambda \text{.}$

		Proof.
	$\square$

Let $(I,{\Omega }^{+})$ be a quiver, let ${\Omega }^{-}$ be the opposite orientation and let ${\Omega }^{\pm }={\Omega }^{+}\cup {\Omega }^{-}\text{.}$ For each $$-\nu =-\sum _{i\in I}{\nu }_i{\alpha }_i\in Q^{-}\text{fix an}\hspace{0.17em}I\text{-graded vector space}V=\underset{i\in I}{⨁}{V}_i,$$ such that $\text{dim}\left(V_i\right)={\nu }_i\text{.}$ Let $$\begin{array}{ccc}{G}_V& =& \prod _{i\in I}GL\left(V_i\right),\\ 𝔤𝔩\left(V\right)& =& \underset{i\in I}{⨁}\text{End}\left(V_i\right),\end{array}\text{and}\begin{array}{ccc}{E}_V& =& \underset{(i\to j)\in {\Omega }^{+}}{⨁}\text{Hom}(V_i,V_j),\\ X_V& =& \underset{(i\to j)\in {\Omega }^{\pm }}{⨁}\text{Hom}(V_i,V_j)\text{.}\end{array}$$ Define a moment map $\mu :X_V\to {𝔤𝔩}_V$ by $$\mu ={\oplus }_{i\in I}{\mu }_i\text{where}{\mu }_i\left(x\right)=\sum _{(i\to j)\in {\Omega }^{\pm }}{x}_{j\to i}{x}_{i\to j}-x_{i\to j}{x}_{j\to i},$$ and let $${\Lambda }_V=\{x\in X_V\hspace{0.17em}|\hspace{0.17em}\mu \left(x\right)=0\hspace{0.17em}\text{and}\hspace{0.17em}x\hspace{0.17em}\text{is nilpotent}\}\text{.}$$ Let $${B\left(\infty \right)}_{-\nu }=\{\text{irreducible components of}\hspace{0.17em}{\Lambda }_V=\}\text{and}B\left(\infty \right)=\underset{-\nu \in Q^{-}}{⨆}{B\left(\infty \right)}_{-\nu }\text{.}$$ For $b\in B$ define $$\text{wt}\left(b\right)=-\nu ,\text{if}\hspace{0.17em}b\in {B\left(\infty \right)}_{-\nu },\text{and}{\epsilon }_i\left(b\right)={\epsilon }_i\left(x\right)=\text{dim}\hspace{0.17em}\text{coker}\left(\left({\oplus }_{j\to i}{V}_j\right)\stackrel{x_{j\to i}}{⟶}{V}_i\right),$$ where $x$ is any generic point of $b\text{.}$

Let $X_0(-\nu )=\{x\in X(-\nu )\hspace{0.17em}|\hspace{0.17em}\mu \left(x\right)=0\}$ and let $X_0^{\prime }(-\nu +r{\alpha }_i,-r{\alpha }_i)$ be the set of triples $(x,\bar{\varphi },\varphi \prime )$ such that $$x\in X_0(-\nu ),\text{and}0\to V(-\nu +r{\alpha }_i)\stackrel{\bar{\varphi }}{⟶}V(-\nu )\stackrel{\varphi \prime }{⟶}V(-r{\alpha }_i)⟶0$$ is an exact sequence of $I\text{-graded}$ vector spaces such that $\text{im}\left(\bar{\varphi }\right)$ is $x\text{-stable.}$ For each $(x,\bar{\varphi },\varphi \prime )\in X_0^{\prime }(-\nu +r{\alpha }_i,-r{\alpha }_i)$ let $$\bar{x}:V(-\nu +r{\alpha }_i)⟶V(-\nu +r{\alpha }_i)\text{and}x\prime :V(-r{\alpha }_i)⟶V(-r{\alpha }_i)$$ be the induced maps, $$\begin{array}{ccccccccc}0& ⟶& {V\left(\nu \right)}_k& \stackrel{{\bar{\varphi }}_k}{⟶}& {V\left(\nu \right)}_k& \stackrel{{\varphi }_k^{\prime }}{⟶}& 0& ⟶& 0\\ & & \phantom{{\bar{x}}_{k\to i}}\downarrow {\bar{x}}_{k\to i}& & \phantom{x_{k\to i}}\downarrow x_{k\to i}& & \phantom{x_{k\to i}^{\prime }}\downarrow x_{k\to i}^{\prime }\\ 0& ⟶& W& \stackrel{{\bar{\varphi }}_i}{⟶}& {V\left(\nu \right)}_i& \stackrel{{\varphi }_i^{\prime }}{⟶}& {ℂ}^r& ⟶& 0\\ & & \phantom{{\bar{x}}_{i\to j}}\downarrow {\bar{x}}_{i\to j}& & \phantom{x_{i\to j}}\downarrow x_{i\to j}& & \phantom{x_{i\to j}^{\prime }}\downarrow x_{i\to j}^{\prime }\\ 0& ⟶& {V\left(\nu \right)}_j& \stackrel{{\bar{\varphi }}_j}{⟶}& {V\left(\nu \right)}_j& \stackrel{{\varphi }_j^{\prime }}{⟶}& 0& ⟶& 0\end{array}$$ The maps $$\begin{array}{ccccccc}{X}_0(-\nu +r{\alpha }_i)& \stackrel{\sim }{⟵}& X_0(-\nu +r{\alpha }_i)\times X_0(-r{\alpha }_i)& \stackrel{\omega }{⟵}& X_0^{\prime }(-\nu +r{\alpha }_i,-r{\alpha }_i)& \stackrel{{\pi }_1}{⟶}& X_0(-\nu )\\ \bar{x}& ⟵& (\bar{x},0)=(\bar{x},x\prime )& ⟵& (x,\bar{\varphi },\varphi \prime )& ⟶& x\end{array}$$ induce an isomorphism $${\stackrel{\sim }{e}}_i^r=\omega \circ {\pi }_1^{-1}:\{b\in {B\left(\infty \right)}_{-\nu }\hspace{0.17em}|\hspace{0.17em}{\epsilon }_i\left(b\right)=r\}\stackrel{\sim }{⟶}\{b\in {B\left(\infty \right)}_{-\nu +r{\alpha }_i}\hspace{0.17em}|\hspace{0.17em}{\epsilon }_i\left(b\right)=0\}\text{.}$$ These maps determine maps ${\stackrel{\sim }{e}}_i$ on $B\left(\infty \right)\text{.}$

This is a realization of the crystal $B\left(\infty \right)\text{.}$

		Proof.
	$\square$

Path crystals

Let $\overrightarrow{B}$ be the set of paths in ${𝔥}_{ℝ}^{*}$ where a path in ${𝔥}_{ℝ}^{*}$ is the image of a piecewise linear map $$\begin{array}{c}p:[0,1]\to {𝔥}_{ℝ}^{*}\text{such that}\hspace{0.17em}p\left(0\right)=0\hspace{0.17em}\text{and}\hspace{0.17em}p\left(1\right)\in P\text{.}\\ \text{PICTURE}\end{array}$$ Define functions $$\begin{array}{ccc}\multicolumn{3}{c}{\text{wt}:\overrightarrow{B}\to P,}\\ {\epsilon }_i\left(p\right):\overrightarrow{B}\to {\mathbb{Z}}_{\ge 0}& \text{and}& {\varphi }_i:\overrightarrow{B}\to {\mathbb{Z}}_{\ge 0}\\ {\stackrel{\sim }{e}}_i:\overrightarrow{B}\to \overrightarrow{B}\cup \left\{0\right\}& \text{and}& {\stackrel{\sim }{f}}_i:\overrightarrow{B}\to \overrightarrow{B}\cup \left\{0\right\}\end{array}$$ by $$\begin{array}{ccc}\text{wt}\left(p\right)& =& p\left(1\right)\\ & =& \text{the endpoint of}\hspace{0.17em}p,\\ {\epsilon }_i\left(p\right)& =& \left|\hspace{0.17em}\lfloor \text{min}\left\{⟨p\left(t\right),{\alpha }_i^{\vee }⟩\hspace{0.17em}\right|\hspace{0.17em}0\le t\le 1\}\rfloor \hspace{0.17em}\right|\\ & =& \text{distance from the most negative point to}\hspace{0.17em}{H}_{{\alpha }_i},\\ {\phi }_i\left(p\right)& =& \left|\hspace{0.17em}\lfloor ⟨p\left(1\right),{\alpha }_i^{\vee }⟩-\text{min}\left\{⟨p\left(t\right),{\alpha }_i^{\vee }⟩\hspace{0.17em}\right|\hspace{0.17em}0\le t\le 1\}\rfloor \hspace{0.17em}\right|\\ & =& \text{distance from the most negative point to}\hspace{0.17em}p\left(1\right),\\ {\stackrel{\sim }{e}}_i\left(p\right)& =& \{\begin{array}{cc}t\mapsto p\left(t\right)+r_i\left(t\right){\alpha }_i,& \text{if}\hspace{0.17em}{r}_i\left(0\right)=0,\\ 0,& \text{otherwise,}\end{array}\\ {\stackrel{\sim }{f}}_i\left(p\right)& =& \{\begin{array}{cc}t\mapsto p\left(t\right)-{\ell }_i\left(t\right){\alpha }_i,& \text{if}\hspace{0.17em}{\ell }_i\left(1\right)=1,\\ 0,& \text{otherwise,}\end{array}\end{array}$$ where $r_i:[0,1]\to [0,1]$ and ${\ell }_i:[0,1]\to [0,1]$ are the monotone functions given by $$\begin{array}{ccc}{r}_i\left(t\right)& =& 1-\text{min}\{1,⟨p\left(s\right),{\alpha }_i^{\vee }⟩-{\epsilon }_i\left(p\right)\hspace{0.17em}|\hspace{0.17em}0\le s\le t\}\\ {\ell }_i\left(t\right)& =& \text{min}\{1,⟨p\left(s\right),{\alpha }_i^{\vee }⟩-{\epsilon }_i\left(p\right)\hspace{0.17em}|\hspace{0.17em}t\le s\le 1\}\end{array}$$ To visualize these operations note that $$\begin{array}{c}⟨p\left(t\right),{\alpha }_i^{\vee }⟩\hspace{0.17em}\text{is the “distance of the point}\hspace{0.17em}p\left(t\right)\hspace{0.17em}\text{from the hyperplane}\hspace{0.17em}{H}_{{\alpha }_i}\text{.”}\\ \text{PICTURE}\end{array}$$ For a path “traveling back and forth with respect to $H_{{\alpha }_i}\text{”,}$ place a hyperplane $H_{\text{mnp}}$ parallel to $H_{{\alpha }_i}$ and through the most negative point of $p\text{.}$ Draw another parallel hyperplane $H_{\text{mnp}+1}$ one unit in the positive direction from $H_{\text{mnp}}$ (i.e. $⟨x,{\alpha }_i^{\vee }⟩=⟨y,{\alpha }_i^{\vee }⟩+1$ for $x\in H_{\text{mnp}},$ $y\in H_{\text{mnp}+1}\text{).}$ Water poured down the tube created by $H_{\text{mnp}}$ and $H_{\text{mnp}+1}$ will create a waterfall and wet those parts of $p$ corresponding to where the function ${\ell }_i$ is increasing. $$\text{PICTURE}$$ The new path ${\stackrel{\sim }{f}}_{i}p$ is the path which follows the same trajectory as $p$ except that the “wet parts” are “reflected with respect to the ${\alpha }_i$ direction”. In the case, figure 2, where the “the water flows in the positive direction” then ${\stackrel{\sim }{f}}_{i}p=0\text{.}$

The set $\overrightarrow{B}$ is a normal crystal.

		Proof.
	$\square$

Define the concatenation or tensor product of paths $p_1$ and $p_2$ to be the path $p_1\otimes p_2$ given by $$(p_1\otimes p_2)\left(t\right)=\{\begin{array}{cc}{p}_1\left(2t\right),& 0\le t\le 1/2,\\ p_1\left(1\right)+p_2(2t-1),& 1/2\le t\le 1\text{.}\end{array}$$ The reverse of the path $p$ is the path $p^{*}$ given by $$p^{*}\left(t\right)=p(1-t),0\le t\le 1\text{.}$$ For $k\in {\mathbb{Z}}_{\ge 0}$ the $k\text{-stretch}$ of $p$ is the path $kp$ given by $$\left(kp\right)\left(t\right)=p\left(kt\right),0\le t\le 1\text{.}$$ Let $C$ be the dominant chamber, let $\rho$ be the half-sum of the positive roots and set $$\begin{array}{c}C-\rho =\{x-\rho \in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}x\in C\}\text{.}\\ \text{PICTURE}\end{array}$$ Write $p\subseteq C-\rho$ if $p\left(t\right)\in C-\rho$ for all $0\le t\le 1\text{.}$ The definitions imply that $$p\in \overrightarrow{B}\hspace{0.17em}\text{is a highest weight path}\text{if and only if}p\subseteq C-\rho \text{.}$$

Let $B$ be a subcrystal of $\overrightarrow{B}$ such that $B_{\mu }$ is finite for all $\mu \in P\text{.}$ Then $${\chi }^B=\sum _{\underset{b\subseteq C-\rho }{b\in B}}{s}_{\text{wt}\left(b\right)},$$ where $s_{\lambda }$ denotes the Weyl character corresponding to $\lambda \in P^{+}\text{.}$

		Proof.
	Let $\mu \in P^{+}\text{.}$ Then $$\begin{array}{cc}{\chi }^B{a}_{\rho }{|}_{a_{\mu +\rho }}=\left(\sum _{w\in W}{(-1)}^{\ell \left(w\right)}{e}^{w\rho }\right)\left(\sum _{p\in B}{e}^{\text{wt}\left(p\right)}\right){|}_{a_{\mu +\rho }}=\sum _{\underset{p\in B}{w\in W}}{(-1)}^{\ell \left(w\right)}{e}^{\text{wt}\left(p\right)+w\rho }{|}_{e^{\mu +\rho }}\text{.}& \text{(2.15)}\end{array}$$ Let $p\in B$ and $w\in W$ be such that $\text{wt}\left(p\right)+w\rho =\mu +\rho \text{.}$ Let $t_0$ be maximal such that there is an $i\in I$ with $w\rho +p\left(t_0\right)\in H_{{\alpha }_i}\text{.}$ If $t_0$ does not exist then $p\in C-\rho$ and $w=1\text{.}$ If $t_0$ does exist set $$\Phi \left(p\right)=\{\begin{array}{cc}{\stackrel{\sim }{f}}_i^{-⟨w\rho ,{\alpha }_i^{\vee }⟩}p,& \text{if}\hspace{0.17em}⟨w\rho ,{\alpha }_i^{\vee }⟩<0,\\ {\stackrel{\sim }{e}}_i^{⟨w\rho ,{\alpha }_i^{\vee }⟩}p,& \text{if}\hspace{0.17em}⟨w\rho ,{\alpha }_i^{\vee }⟩>0\text{.}\end{array}$$ Then $\text{wt}\left(p\right)+w\rho =\text{wt}\left(\Phi \left(p\right)\right)+s_{i}w\rho$ and the pairs $(p,w)$ and $(\Phi \left(p\right),s_{i}w)$ cancel in the sum (???). $\square$

Let $\lambda \in P^{+}$ and let $b_{\lambda }^{+}$ be a highest weight path with $\text{wt}\left(b_{\lambda }^{+}\right)=\lambda \text{.}$ For example, $b_{\lambda }^{+}$ might be the path given by $$b_{\lambda }^{+}\left(t\right)=t\lambda ,0\le t\le 1\text{.}$$ Define $$B\left(\lambda \right)=\left\{{\stackrel{\sim }{f}}_{i_k}\cdots {\stackrel{\sim }{f}}_{i_2}{\stackrel{\sim }{f}}_{i_1}{b}_{\lambda }^{+}\hspace{0.17em}\right|\hspace{0.17em}1\le i_1,\dots ,i_k\le n,k\in {\mathbb{Z}}_{\ge 0}\}$$ so that $b\left(\lambda \right)$ is the collection of paths obtained by applying finite sequences of ${\stackrel{\sim }{f}}_i$ to $b_{\lambda }^{+}\text{.}$

Let $\lambda \in P^{+}\text{.}$

(a)	$s_{\lambda }={\displaystyle \sum _{b\in B\left(\lambda \right)}}{e}^{\text{wt}\left(b\right)}\text{.}$
(b)	$s_{\lambda }\left(q^{\rho }\right)={\displaystyle \prod _{\alpha \in R^{+}}}\frac{\left[⟨\lambda +\rho ,{\alpha }^{\vee }⟩\right]}{\left[⟨\rho ,{\alpha }^{\vee }⟩\right]}={\displaystyle \sum _{b\in B\left(\lambda \right)}}{q}^{⟨\text{wt}\left(b\right),\rho ⟩}$
(c)	$s_{\lambda }\left(1\right)={\displaystyle \prod _{\alpha \in R^{+}}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}=\text{Card}\left(B\left(\lambda \right)\right)\text{.}$
(d)	If $K_{\lambda \mu }$ are defined by $s_{\lambda }={\displaystyle \sum _{\mu \in P}}{K}_{\lambda \mu }{m}_{\mu }$ then $K_{\lambda \mu }=\text{Card}\left({B\left(\lambda \right)}_{\mu }\right),$

		Proof.
	$\square$

Let $J\subseteq I\text{.}$ The crystal $${\text{Res}}_{J}B\text{is}B\text{with only those crystal operators}{\stackrel{\sim }{e}}_j,{\stackrel{\sim }{f}}_j\text{for}\hspace{0.17em}j\in J\text{.}$$ This is a crystal for the parabolic subsystem $(W_J,C_J,P)\text{.}$

Let $\lambda \in P^{+}$ and let $B\left(\lambda \right)$ be the irreducible highest weight crystal of highest weight $\lambda \text{.}$ $${\text{Res}}_{J}B\left(\lambda \right)=\underset{\underset{p\subseteq C_J-{\rho }_J}{p\in B\left(\lambda \right)}}{⨁}B\left(\text{wt}\left(p\right)\right)B\left(\lambda \right)\otimes B\left(\mu \right)=\underset{\underset{p_{\lambda }\otimes p\subseteq C-\rho }{p\in B\left(\mu \right)}}{⨁}B\left(\text{wt}\left(p\right)\right){B\left(\lambda \right)}^{*}\cong B(-w_0\lambda )\text{.}$$

		Proof.
	$\square$

Let $\mu ,\nu ,\lambda \in P^{+}$ and let $\tau \in P_J^{+}\text{.}$ $$\begin{array}{ccc}{c}_{\mu \nu }^{\lambda }& =& \text{Card}\left(\{b\in B\left(\mu \right)\hspace{0.17em}|\hspace{0.17em}\text{wt}(b_{\mu }^{+}\otimes b)=\lambda \text{and}{b}_{\mu }^{+}\otimes b\in C-\rho \}\right),\text{and}\\ c_{J,\tau }^{\lambda }& =& \text{Card}\left(\{b\in B\left(\lambda \right)\hspace{0.17em}|\hspace{0.17em}{\text{wt}}_J\left(b\right)=\tau \text{and}b\subseteq C_J-{\rho }_J\}\right)\text{.}\end{array}$$ Then $$s_{\mu }{s}_{\nu }=\sum _{\lambda }{c}_{\mu \nu }^{\lambda }{s}_{\lambda }\text{and}{s}_{\lambda }=\sum _{\tau \in P_J^{+}}{c}_{J,\tau }^{\lambda }{s}_{\tau }^J\text{.}$$

		Proof.
	$\square$

For paths $p_1,p_2\in \overrightarrow{B}$ define $$d(p_1,p_2)=\text{max}\left\{|p_1\left(t\right)-p_2\left(t\right)|\hspace{0.17em}\right|\hspace{0.17em}0\le t\le 1\}\text{.}$$

The operators $${\stackrel{\sim }{e}}_i:\overrightarrow{B}\to \overrightarrow{B}\cup \left\{0\right\}\text{and}{\stackrel{\sim }{f}}_i:\overrightarrow{B}\to \overrightarrow{B}\cup \left\{0\right\}$$ are the unique operators such that

(1)	$$\begin{array}{ccc}{\stackrel{\sim }{e}}_i(b_1\otimes b_2)& =& \{\begin{array}{cc}{\stackrel{\sim }{e}}_i{b}_1\otimes b_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)\ge {\epsilon }_i\left(b_2\right),\\ b_1\otimes {\stackrel{\sim }{e}}_i{b}_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)<{\epsilon }_i\left(b_2\right),\end{array}\text{and}\\ {\stackrel{\sim }{f}}_i(b_1\otimes b_2)& =& \{\begin{array}{cc}{\stackrel{\sim }{f}}_i{b}_1\otimes b_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)>{\epsilon }_i\left(b_2\right),\\ b_1\otimes {\stackrel{\sim }{f}}_i{b}_2,& \text{if}\hspace{0.17em}{\phi }_i\left(b_1\right)\le {\epsilon }_i\left(b_2\right),\end{array}\end{array}$$
(2)	The operators $$s_{i}p=\{\begin{array}{cc}{\stackrel{\sim }{f}}_i^{⟨\text{wt}\left(p\right),{\alpha }_i^{\vee }⟩}p,& \text{if}\hspace{0.17em}⟨\text{wt}\left(p\right),{\alpha }_i^{\vee }⟩>0,\\ p,& \text{if}\hspace{0.17em}⟨\text{wt}\left(p\right),{\alpha }_i^{\vee }⟩=0,\\ {\stackrel{\sim }{e}}_i^{⟨\text{wt}\left(p\right),{\alpha }_i^{\vee }⟩}p,& \text{if}\hspace{0.17em}⟨\text{wt}\left(p\right),{\alpha }_i^{\vee }⟩<0,\end{array}$$ define an action of $W$ on $\overrightarrow{B}\text{.}$
(3)	${\epsilon }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{e}}_i^{k}b\ne 0\}$ and ${\phi }_i\left(b\right)=\text{max}\left\{k\hspace{0.17em}\right|\hspace{0.17em}{\stackrel{\sim }{f}}_i^{k}b\ne 0\}\text{.}$
(4)	If ${\stackrel{\sim }{e}}_i\left(b\right)\ne 0$ then ${\stackrel{\sim }{f}}_i{\stackrel{\sim }{e}}_{i}b=b$ and ${\left({\stackrel{\sim }{e}}_{i}b\right)}^{*}={\stackrel{\sim }{f}}_i{b}^{*}\text{.}$ If ${\stackrel{\sim }{f}}_i\left(b\right)\ne 0$ then ${\stackrel{\sim }{e}}_i{\stackrel{\sim }{f}}_{i}b=b$ and ${\left({\stackrel{\sim }{f}}_{i}b\right)}^{*}={\stackrel{\sim }{e}}_i{b}^{*}\text{.}$
(5)	If $b\in \overrightarrow{B}$ and $k\in {\mathbb{Z}}_{>0}$ then $$k\left({\stackrel{\sim }{e}}_{i}b\right)={\stackrel{\sim }{e}}_i^k\left(kb\right)\text{and}k\left({\stackrel{\sim }{f}}_{i}b\right)={\stackrel{\sim }{f}}_i^k\left(kb\right)\text{.}$$
(6)	There is a constant $c,$ depending only on $(W,C,P)$ such that If ${\stackrel{\sim }{e}}_i{b}_1\ne 0$ and ${\stackrel{\sim }{e}}_i{b}_2\ne 0$ then $d({\stackrel{\sim }{e}}_i{b}_1,{\stackrel{\sim }{e}}_i{b}_2)<c·d(b_1,b_2),$ and If ${\stackrel{\sim }{f}}_i{b}_1\ne 0$ and ${\stackrel{\sim }{f}}_i{b}_2\ne 0$ then $d({\stackrel{\sim }{f}}_i{b}_1,{\stackrel{\sim }{f}}_i{b}_2)<c·d(b_1,b_2)\text{.}$

		Proof.
	$\square$

The crystal $B_n^{\text{<mo>⊗</mo><mi>k</mi>}}$

Let $$\begin{array}{ccc}{ℝ}^n& =& \sum _{i=1}^nℝ{\epsilon }_i,\text{with the}\hspace{0.17em}{\epsilon }_i\hspace{0.17em}\text{an orthonormal basis,}\\ W& =& S_n,\text{acting on}\hspace{0.17em}{ℝ}^n\hspace{0.17em}\text{by permuting the}\hspace{0.17em}{\epsilon }_i,\\ C& =& \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_1<{\mu }_2<\cdots <{\mu }_n\},\\ L& =& \sum _{i=1}^n\mathbb{Z}{\epsilon }_i,\end{array}$$ and let $b_i$ be the straightline path from $0$ to ${\epsilon }_i\text{.}$ Then $$B_n^{\text{<mo>⊗</mo><mi>k</mi>}}=\left\{b_{i_1}\cdots b_{i_k}\hspace{0.17em}\right|\hspace{0.17em}1\le i_1,\dots ,i_k\le n\}$$ is the set of length $k$ paths in ${ℝ}^n$ where each step is a unit step in one of the directions ${\epsilon }_1,\dots ,{\epsilon }_n\text{.}$ To compute the operators ${\stackrel{\sim }{e}}_i$ and ${\stackrel{\sim }{f}}_i$ on a path $b=b_{i_1}\cdots b_{i_k}$ in $B_n^{\otimes k}$ place the value $$\begin{array}{ccc}+1& =& ⟨{\epsilon }_i,{\alpha }_i^{\vee }⟩\hspace{0.17em}\text{over each}\hspace{0.17em}{b}_i,\\ -1& =& ⟨{\epsilon }_{i+1},{\alpha }_i^{\vee }⟩\hspace{0.17em}\text{over each}\hspace{0.17em}{b}_{i+1},\\ 0& =& ⟨{\epsilon }_j,{\alpha }_i^{\vee }⟩\hspace{0.17em}\text{over each}\hspace{0.17em}{b}_j,\hspace{0.17em}j\ne i,i+1,\end{array}$$ Ignoring $0\text{s}$ read this sequence of $\pm 1$ from left to right and succesively remove adjacent $(+1,-1)$ pairs until the sequence is of the form $$\begin{array}{cc}\text{cogood}& \text{good}\\ \downarrow & \downarrow \\ \underset{\underset{\text{conormal nodes}}{⏟}}{+1\hspace{0.17em}+1\hspace{0.17em}\dots +1}& \underset{\underset{\text{normal nodes}}{⏟}}{-1\hspace{0.17em}-1\hspace{0.17em}\dots -1}\end{array}$$ The $-1\text{s}$ in this sequence are the normal nodes and the $+1\text{s}$ are the conormal nodes. The good node is the leftmost normal node and the cogood node is the right most conormal node. The good node is exactly at the position where the path $b$ is at its most negative point with respect to the hyperplane $H_{{\alpha }_i}\text{.}$ Then $$\begin{array}{ccc}{\epsilon }_i\left(b\right)& =& \text{(\# of normal nodes),}\\ {\phi }_i\left(b\right)& =& \text{(\# of conormal nodes),}\\ {\stackrel{\sim }{e}}_i\left(b\right)& =& \text{same as}\hspace{0.17em}b\hspace{0.17em}\text{except with the cogood node path step changed to}\hspace{0.17em}{b}_i,\\ {\stackrel{\sim }{f}}_i\left(b\right)& =& \text{same as}\hspace{0.17em}b\hspace{0.17em}\text{except with the good node path step changed to}\hspace{0.17em}{b}_i\text{.}\end{array}$$ For example, if $n=5$ and $k=30$ and $$b=b_4{b}_3{b}_2{b}_1{b}_2{b}_2{b}_4{b}_4{b}_1{b}_2{b}_3{b}_3{b}_2{b}_1{b}_1{b}_2{b}_3{b}_3{b}_2{b}_1{b}_4{b}_5{b}_5{b}_1{b}_1{b}_1{b}_1{b}_2{b}_2{b}_4$$ then the parentheses in the table $$\begin{array}{ccccccccccccccccccccccc}& & & (& )& -1& & & (& )& -1& & & (& (& )& & & )& +1& & & \\ 0& 0& 0& +1& -1& -1& 0& 0& +1& -1& 0& 0& -1& +1& +1& -1& 0& 0& -1& +1& 0& 0& 0\\ b_4& b_3& b_3& b_1& b_2& b_2& b_4& b_4& b_1& b_2& b_3& b_3& b_2& b_1& b_1& b_2& b_3& b_3& b_2& b_1& b_4& b_5& b_5\end{array}$$ indicate the $(+1,-1)$ pairings and the numbers in the top row indicate the resulting sequence of $-1\text{s}$ and $+1\text{s.}$ Then $$\begin{array}{ccc}{\epsilon }_1\left(b\right)& =& 2,{\phi }_1\left(b\right)=3,\\ {\stackrel{\sim }{e}}_i\left(b\right)& =& b_4{b}_3{b}_3{b}_1{b}_2{b}_2{b}_4{b}_4{b}_1{b}_2{b}_3{b}_3{b}_2{b}_1{b}_1{b}_2{b}_3{b}_3{b}_2\underset{\_}{b_2}{b}_4{b}_5{b}_5{b}_1{b}_1{b}_1{b}_1{b}_2{b}_2{b}_2{b}_4,\\ {\stackrel{\sim }{f}}_i\left(b\right)& =& b_4{b}_3{b}_3{b}_1{b}_2{b}_2{b}_4{b}_4{b}_1{b}_2{b}_3{b}_3\underset{\_}{b_1}{b}_1{b}_1{b}_2{b}_3{b}_3{b}_2{b}_1{b}_4{b}_5{b}_5{b}_1{b}_1{b}_1{b}_1{b}_2{b}_2{b}_2{b}_4,\end{array}$$

The highest weight paths in $B_n^{\otimes k}$ are the $b=b_{i_1}\cdots b_{i_k}$ such that for every $1\le j\le k$ and every $1\le i\le n$ the $$\left(\text{\# of}\hspace{0.17em}{b}_i\hspace{0.17em}\text{in}\hspace{0.17em}{b}_{i_1}\cdots b_{i_j}\right)\ge \left(\text{\# of}\hspace{0.17em}{b}_{i+1}\hspace{0.17em}\text{in}\hspace{0.17em}{b}_{i_1}\cdots b_j\right)\text{.}$$ The map $$Q:\left\{\text{highest weight paths in}\hspace{0.17em}{B}_n^{\otimes k}\right\}\stackrel{1-1}{⟷}\{\text{standard tableaux with}\hspace{0.17em}k\hspace{0.17em}\text{boxes and}\hspace{0.17em}\le n\hspace{0.17em}\text{rows}\}$$ is given by making the standard tableau $$Q\left(b\right)\text{such that}\text{entry}\hspace{0.17em}j\hspace{0.17em}\text{is in row}\hspace{0.17em}i\hspace{0.17em}\text{if}\hspace{0.17em}b\hspace{0.17em}\text{has}\hspace{0.17em}{b}_i\hspace{0.17em}\text{in position}\hspace{0.17em}j\hspace{0.17em}\text{(i.e. if}\hspace{0.17em}{b}_{i_j}=b_i\text{).}$$ For example, if $k=n=4$ then $B_4=\{b_1,b_2,b_3,b_4\}$ and the map $Q$ is given by $$\begin{array}{ccc}\begin{array}{ccc}{b}_1{b}_1{b}_1{b}_1& ⟼& 1234\end{array}& & \begin{array}{ccc}{b}_1{b}_2{b}_3{b}_4& ⟼& 1\\ & & 2\\ & & 3\\ & & 4\end{array}\\ \\ \begin{array}{ccc}{b}_1{b}_1{b}_1{b}_2& ⟼& 123\\ & & 4\end{array}& & \begin{array}{ccc}{b}_1{b}_1{b}_2{b}_3& ⟼& 12\\ & & 3\\ & & 4\end{array}\\ \\ \begin{array}{ccc}{b}_1{b}_1{b}_2{b}_1& ⟼& 124\\ & & 3\end{array}& & \begin{array}{ccc}{b}_1{b}_2{b}_1{b}_3& ⟼& 13\\ & & 2\\ & & 4\end{array}\\ \\ \begin{array}{ccc}{b}_1{b}_2{b}_1{b}_1& ⟼& 134\\ & & 2\end{array}& & \begin{array}{ccc}{b}_1{b}_2{b}_3{b}_1& ⟼& 14\\ & & 2\\ & & 3\end{array}\\ \\ \begin{array}{ccc}{b}_1{b}_1{b}_2{b}_2& ⟼& 12\\ & & 34\end{array}& & \begin{array}{ccc}{b}_1{b}_2{b}_1{b}_2& ⟼& 12\\ & & 13\\ & & 24\end{array}\end{array}$$ Let $x_i=e^{{\epsilon }_i}\text{.}$ For a highest weight path $b\in B_n^{\otimes k}$ $$\text{wt}\left(b\right)={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n\text{if the shape of}\hspace{0.17em}Q\left(b\right)\hspace{0.17em}\text{is}\lambda =({\lambda }_1,\dots ,{\lambda }_n)$$ and so the character of the crystal $B_n^{\otimes k}$ is $${(x_1+\cdots +x_n)}^k=\sum _{\underset{\ell \left(\lambda \right)\le n}{\lambda ⊢n}}{f}^{\lambda }{s}_{\lambda },\text{where}{f}^{\lambda }=\text{(\# of standard tableaux of shape}\hspace{0.17em}\lambda \text{).}$$

Multisegments

$$\begin{array}{cc}{A}_{n-1}& \begin{array}{c} 2 3 n-1 n \end{array}\\ A_{\infty }& \begin{array}{c} -2 -1 0 1 2 \end{array}\\ A_{n-1}^{\left(1\right)}& \begin{array}{c} 2 3 0 n-1 n \end{array}\end{array}$$ Let $$I=\{\begin{array}{cc}\{1,2,\dots ,\ell -1\},& \text{in type}\hspace{0.17em}{A}_{\ell -1}\\ \mathbb{Z},& \text{in type}\hspace{0.17em}{A}_{\infty }\\ \mathbb{Z}/\ell \mathbb{Z},& \text{in type}\hspace{0.17em}{A}_{\ell -1}^{\left(1\right)}\end{array}$$ The elements of $I$ index the (isomorphism classes) of simple representations of the quiver.

A segment is a row of boxes on graph paper with diagonals indexed by $\mathbb{Z}$ $$[i,d)=[i,i+d-1]=(d;i+d-1]$$ to denote a row of boxes of length $d$ such that the leftmost box has content $i$ and the rightmost box has content $i+d-1\text{.}$ The set of segments is $$R^{+}=\{\begin{array}{cc}\left\{[i;d)\hspace{0.17em}\right|\hspace{0.17em}i\in I,1\le d\le \ell -i\},& \text{in type}\hspace{0.17em}{A}_{\ell -1},\\ \left\{[i;d)\hspace{0.17em}\right|\hspace{0.17em}i\in I,d\in {\mathbb{Z}}_{\ge 0}\}& \text{in type}\hspace{0.17em}{A}_{\infty },\\ \left\{[i;d)\hspace{0.17em}\right|\hspace{0.17em}i\in I,d\in \mathbb{Z}/\ell \mathbb{Z}\}& \text{in type}\hspace{0.17em}{A}_{\ell -1}^{\left(1\right)},\end{array}$$ The elements of $R^{+}$ index the (isomorphism classes) of indecomposable (nilpotent) representations of the quiver. A multisegment is a collection of segment, i.e. an element of $$\stackrel{\sim }{B}\left(\infty \right)=\sum _{\alpha \in R^{+}}{\mathbb{Z}}_{\ge 0}\alpha$$ A multisegment is aperiodic if it does not contain $$[0;d)+[1;d)+\cdots +[\ell -1;d),\text{for any}\hspace{0.17em}d\in {\mathbb{Z}}_{>0}\text{.}$$ Pictorially, a multisegment is aperiodic if it does not contain a box of height $\ell \text{.}$ Let $$B\left(\infty \right)=\left\{\text{aperiodic multisegments}\right\}\text{.}$$ In types $A_{\ell -1}$ and $A_{\infty },$ $B\left(\infty \right)=\stackrel{\sim }{B}\left(\infty \right)\text{.}$ The elements of $B\left(\infty \right)$ index the isomorphism classes of nilpotent representations of the quiver.

The partial order

Consider graph paper with diagonal indexed by $\mathbb{Z}\text{.}$ A segment is a row $$[i,j]=\text{PICTURE}$$ of boxes. A multisegment is a collection of rows of boxes. $$\text{PICTURE}$$ Alternatively a multisegment $\lambda$ can be viewed as a function $$\lambda :\left\{\text{segments}\right\}⟶{\mathbb{Z}}_{\ge 0}\text{where}\lambda \left([i,j]\right)=\text{(\# of rows}\hspace{0.17em}[i,j]\hspace{0.17em}\text{in}\hspace{0.17em}\lambda \text{).}$$ The set of segments is ordered by inclusion. $$\text{PICTURE}$$ Define $$\lambda (\supseteq [i,j])=\sum _{[r,s]\supseteq [i,j]}\lambda \left([r,s]\right)\text{.}$$ Then $$\begin{array}{c}\lambda \left([i,j]\right)=\lambda (\supseteq [i-1,j+1])-\lambda (\supseteq [i-1,j])-\lambda (\supseteq [i,j+1])+\lambda (\supseteq [i,j])\text{.}\\ \text{PICTURE}\end{array}$$ and so the multisegment $\lambda$ can be specified by the numbers $\lambda (\supseteq [i,j])\text{.}$ Note that $$\lambda (\supseteq \left[i\right])=\text{(\# of boxes in}\hspace{0.17em}\lambda \hspace{0.17em}\text{in diagonal}\hspace{0.17em}i\text{).}$$ Define a partial order on multisegments by $$\lambda \ge \mu \text{if}\lambda (\supseteq [i,j])\ge \mu (\supseteq [i,j])\text{for all segments}\hspace{0.17em}[i,j]\text{.}$$

If $[b,c]\supseteq [a,d]$ are segments define a degeneration $$R_{[b,c],[a,d]}:\left\{\text{multisegments}\right\}⟶\left\{\text{multisegments}\right\}$$ by $$\begin{array}{ccc}{R}_{[b,c],[a,d]}\lambda \left([a,d]\right)& =& \lambda \left([a,d]\right)-1,\\ R_{[b,c],[a,d]}\lambda \left([b,d]\right)& =& \lambda \left([b,d]\right)+1,\\ R_{[b,c],[a,d]}\lambda \left([a,c]\right)& =& \lambda \left([a,c]\right)+1,\\ R_{[b,c],[a,d]}\lambda \left([b,c]\right)& =& \lambda \left([b,c]\right)-1,\\ R_{[b,c],[a,d]}\lambda \left([i,j]\right)& =& \lambda \left([i,j]\right),\text{if}\hspace{0.17em}[i,j]\ne [a,d],[a,c],[b,d],[b,c]\text{.}\end{array}$$ The degeneration $R_{[b,c],[a,d]}\lambda$ is elementary if $$\lambda \left([i,j]\right)=0\text{for all}\hspace{0.17em}[b,c]\subseteq [i,j]\subseteq [a,d]\hspace{0.17em}\text{except}\hspace{0.17em}[i,j]=[b,c],[a,c],[b,d]\hspace{0.17em}\text{or}\hspace{0.17em}[a,d]\text{.}$$ Pictorially a degeneration takes $$\begin{array}{ccc}\text{PICTURE}& ⟶& \text{PICTURE}\end{array}$$ and $$\text{PICTURE}⟶\text{PICTURE}\text{for}\hspace{0.17em}c=b-1,$$ or, equivalently, $$\begin{array}{ccc}\text{PICTURE}& ⟶& \text{PICTURE}\end{array}$$

Let $A_{\infty }$ be the quiver $(I,{\Omega }^{+})$ with $$I=\mathbb{Z},{\Omega }^{+}=\{i\to i+1\hspace{0.17em}|\hspace{0.17em}i\in \mathbb{Z}\}\text{.}$$ Fix an $I\text{-graded}$ vector space $$\begin{array}{ccc}V& =& \underset{i\in I}{⨁}{V}_i,\\ \text{and let}{E}_V& =& \underset{i\to i+1}{⨁}\text{Hom}(V_i,V_{i+1}),\\ {GL}_V& =& \prod _{i\in \mathbb{Z}}GL\left(V_i\right),\text{which acts on}\hspace{0.17em}{E}_V\text{, and}\\ {𝒩}_V& =& \{x\in E_V\hspace{0.17em}|\hspace{0.17em}x\hspace{0.17em}\text{is a nilpotent element of Hom}(V,V)\}\text{.}\end{array}$$ The map $$\begin{array}{c}\begin{array}{ccc}{𝒩}_V& ⟶& \left\{\text{multisegments}\right\}\\ x& ⟼& {\lambda }_x\end{array}\text{given by}\\ \lambda (\supseteq \left[i\right])=\text{dim}\left(V_i\right)\text{and}\lambda (\supseteq [i,j])=\text{rank}(\lambda :V_i\to \cdots \to V_j)\end{array}$$ provides a bijection $$\left\{\text{multisegments}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}\lambda (\supseteq \left[i\right])=\text{dim}\left(V_i\right)\}⟷\left\{{GL}_V\hspace{0.17em}\text{orbits in}\hspace{0.17em}{𝒩}_V\right\}$$

Let $\lambda$ and $\mu$ be multisegments and let ${𝕆}_{\lambda }$ and ${𝕆}_{\mu }$ be the corresponding orbits in ${𝒩}_V/{GL}_V\text{.}$ Then the following are equivalent

(1)	$\lambda \ge \mu ,$
(2)	$\overline{{𝕆}_{\lambda }}\supseteq {𝕆}_{\mu },$
(3)	$\lambda =R_{i_1}\cdots R_{i_r}\mu$ for some sequence of elementary degenerations $R_{i_1},\dots ,R_{i_r}\text{.}$

		Proof.
	(1) $⟹$ (2): $$\text{PICTURE}+\epsilon \text{PICTURE}\cong \text{PICTURE},$$ and so $${𝕆}_{\text{PICTURE}}\subseteq \overline{{𝕆}_{\text{PICTURE}}}\text{.}$$ (2) $⟹$ (3): If ${𝕆}_{\mu }\subseteq \overline{{𝕆}_{\lambda }}$ then $$\mu (\supseteq [i,j])=\text{rank}(\mu :V_i\to \cdots \to V_j)\le rank(\lambda :V_i\to \cdots \to V_j)=\lambda (\supseteq [i,j])\text{.}$$ (3) $⟹$ (1): Assume $\lambda (\supseteq [i,j])\ge \mu (\supseteq [i,j])$ for all segments $[i,j]\text{.}$ Find a sequence $R_{i_1}\cdots R_{i_r}$ of elementary degenerations which takes $\mu$ to $\lambda ,$ i.e. $$R_{i_1}\cdots R_{i_r}\mu =\lambda \text{.}$$ $\square$

The crystal graph

Let $$\lambda =\left[\begin{array}{cccc}{(\lambda +\rho )}_1& {(\lambda +\rho )}_2& \cdots & {(\lambda +\rho )}_n\\ {(\mu +\rho )}_1& {(\mu +\rho )}_2& \cdots & {(\mu +\rho )}_n\end{array}\right]=\left(\begin{array}{cccc}{(\lambda +\rho )}_1& {(\lambda +\rho )}_2& \cdots & {(\lambda +\rho )}_n\\ d_1& d_2& \cdots & d_n\end{array}\right]$$ be a multisegment and assume that it is ordered so that

(a)	${(\lambda +\rho )}_i\ge {(\lambda +\rho )}_{i+1},$
(b)	${(\mu +\rho )}_i\le {(\mu +\rho )}_{i+1}$ if ${(\lambda +\rho )}_i={(\lambda +\rho )}_{i+1},$

These conditions are equivalent to saying that

$\text{(a}\prime \text{)}$	The $𝔤𝔩\left(n\right)\text{-weight}$ $\lambda$ is integrally dominant,
$\text{(b}\prime \text{)}$	$\mu =w\circ \nu$ where $\nu$ is integrally dominant and $w$ is longest in its coset $W_{\lambda +\rho }{W}_{\mu +\rho }\text{.}$

Place $-1$ above each ${(\lambda +\rho )}_j=i,$
$+1$ above each ${(\lambda +\rho )}_j=i-1,$
$0$ above each ${(\lambda +\rho )}_j\ne i,i+1\text{.}$ Then, ignoring $0\text{s,}$ read the sequence of $+1\text{s,}$ $-1\text{s}$ left to right and successively cancel adjacent $(-1,+1)$ pairs to get a sequence of the form $$\begin{array}{cc}\text{cogood}& \text{good}\\ \downarrow & \downarrow \\ \underset{\underset{\text{conormal nodes}}{⏟}}{+1\hspace{0.17em}+1\hspace{0.17em}\dots +1}& \underset{\underset{\text{normal nodes}}{⏟}}{-1\hspace{0.17em}-1\hspace{0.17em}\dots -1}\end{array}$$ The $-1\text{s}$ in this sequence are the normal nodes and the $+1\text{s}$ are the conormal nodes. The good node is the leftmost normal node and the cogood node is the right most conormal node.

Define $$\begin{array}{c}\text{wt}\left(\lambda \right)=\sum _{i\in I}-\left(\text{number of boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{in}\hspace{0.17em}\lambda \right)\hspace{0.17em}{\alpha }_i,\text{and}\\ {\epsilon }_i\left(\lambda \right)=\left(\text{number of normal nodes}\right),{\phi }_i\left(\lambda \right)=\left(\text{number of conormal nodes}\right),\\ {\stackrel{\sim }{e}}_i\lambda =(\text{same as}\hspace{0.17em}\lambda \hspace{0.17em}\text{but with the good node}\hspace{0.17em}{(\lambda +\rho )}_j=i\hspace{0.17em}\text{changed to}\hspace{0.17em}i-1),\\ {\stackrel{\sim }{f}}_i\lambda =(\text{same as}\hspace{0.17em}\lambda \hspace{0.17em}\text{but with the cogood node}\hspace{0.17em}{(\lambda +\rho )}_j=i-1\hspace{0.17em}\text{changed to}\hspace{0.17em}i),\end{array}$$ for each $i\in I\text{.}$

Remark. If this algorithm is being executed where $I=\mathbb{Z}/\ell \mathbb{Z}$ then take ${(\lambda +\rho )}_j=\ell ,$ when $i=0$ and ${(\lambda +\rho )}_j\equiv 0$ mod $\ell ,$ and
${(\lambda +\rho )}_j=0,$ when $i=1$ and ${(\lambda +\rho )}_j\equiv 0$ mod $\ell \text{.}$

(a)	In type $A_{\ell -1}^{\left(1\right)},$ $B\left(\infty \right)$ is the connected component of $\varnothing$ in the crystal graph $\stackrel{\sim }{B}\left(\infty \right)\text{.}$
(b)	$B\left(\infty \right)$ is the crystal graph of $U_v^{-}𝔤\text{.}$

The crystals $B\left(\Lambda \right)$

Type $A_{\ell -1}\text{:}$ Let $$\lambda =\sum _{i=1}^{\ell }{\lambda }_i{\epsilon }_i=\sum _{i\in I}{\gamma }_i{\omega }_i\in P^{+},$$ and identify $\lambda$ with the partition which has ${\lambda }_i$ boxes in row $i\text{.}$ Let $$B\left(\lambda \right)=\left\{\text{column strict tableaux of shape}\hspace{0.17em}\lambda \right\}$$ and define an imbedding $$\begin{array}{ccc}B\left(\lambda \right)& ⟶& B\left(\infty \right)\\ P& ⟼& \left[\begin{array}{ccccccccccccc}1& 1& \cdots & 1& 2& 2& \cdots & 2& \cdots & n& n& \cdots & n\\ i_1& i_2& \cdots & i_{{\lambda }_1}& i_{{\lambda }_1+1}& & \cdots & i_{{\lambda }_1+{\lambda }_2}& \cdots & & & \cdots & i_k\end{array}\right]\end{array}$$ where the entries $i_1{i}_2\cdots i_k$ are the entries of $P$ read in Arabic reading order.

The tensor product representation

The $\ell \text{-dimensional}$ simple $U_q{𝔰𝔩}_{\ell }\text{-module}$ of highest weight ${\omega }_1$ is given by $$L\left({\omega }_1\right)=ℂ\text{-span}\{v_0,\dots ,v_{\ell -1}\}$$ with $U_q{𝔰𝔩}_l\text{-action}$ $$e_i{v}_j=\{\begin{array}{cc}{v}_{i-1},& \text{if}\hspace{0.17em}j=i,\\ 0,& \text{if}\hspace{0.17em}j\ne i,\end{array}{f}_i{v}_j=\{\begin{array}{cc}{v}_i,& \text{if}\hspace{0.17em}j=i-1,\\ 0,& \text{if}\hspace{0.17em}j\ne i,\end{array}{k}_i{v}_j=\{\begin{array}{cc}q{v}_{i-1},& \text{if}\hspace{0.17em}j=i,\\ q^{-1}{v}_i,& \text{if}\hspace{0.17em}j=i-1,\\ v_j,& \text{if}\hspace{0.17em}j\ne i,i-1\text{.}\end{array}$$ Then $${L\left({\omega }_1\right)}^{\otimes k}=ℂ\text{-span}\{v_{j_1}\otimes \cdots \otimes v_{j_k}\hspace{0.17em}|\hspace{0.17em}1\le j_1,j_2,\dots ,j_k\le \ell \}\text{.}$$ If $v=v_{j_1}\otimes \cdots \otimes v_{j_k}$ place $$\begin{array}{c}+1\hspace{0.17em}\text{over each}\hspace{0.17em}{v}_{i-1}\hspace{0.17em}\text{in}\hspace{0.17em}v,\\ -1\hspace{0.17em}\text{over each}\hspace{0.17em}{v}_i\hspace{0.17em}\text{in}\hspace{0.17em}v,\\ 0\hspace{0.17em}\text{over each}\hspace{0.17em}{v}_j,\hspace{0.17em}j\ne i,i-1\text{.}\end{array}$$ Then the $U_q{𝔰𝔩}_{\ell }\text{-action}$ on ${L\left({\omega }_1\right)}^{\otimes k}$ is given by $$\begin{array}{c}{e}_i\left(v\right)=\sum _{v^{-}}{q}^{-(\text{sum of}\hspace{0.17em}\pm 1\text{s before}\hspace{0.17em}v/v^{-})}{v}^{-},f_i\left(v\right)=\sum _{v^{+}}{q}^{(\text{sum of}\hspace{0.17em}\pm 1\text{s after}\hspace{0.17em}{v}^{+}/v)}{v}^{+},\\ k_i\left(v\right)=q^{(\text{sum of}\hspace{0.17em}\pm 1\text{s for}\hspace{0.17em}v)}v,\end{array}$$ where the first sum is over all $v^{-}$ which are obtained from $v$ by changing a $v_i$ to $v_{i-1}$ and the second sum is over all $v^{+}$ which are obtained from $v$ by changing a $v_{i-1}$ to $v_i\text{.}$

The Fock space

Let $\mu \in {𝔥}^{*}$ for ${𝔤𝔩}_n\text{.}$ Define $${ℱ}_{\mu }=ℂ\text{-span}\{\text{multisegments}\hspace{0.17em}\lambda =\lambda /\mu \}\text{.}$$ Define an action of $U_v{\widehat{𝔰𝔩}}_{\ell }$ on ${ℱ}_{\mu }$ by $$\begin{array}{cc}{e}_i\lambda =\sum _{c(\lambda /{\lambda }^{-})\equiv i}{q}^{(\text{sum of}\hspace{0.17em}\pm 1\text{s before}\hspace{0.17em}\lambda /{\lambda }^{-})}{\lambda }^{-},& f_i\lambda =\sum _{c({\lambda }^{+}/\lambda )\equiv i}{q}^{(\text{sum of}\hspace{0.17em}\pm 1\text{s before}\hspace{0.17em}{\lambda }^{+}/\lambda )}{\lambda }^{+},\\ k_i\lambda =q^{(\text{sum of}\hspace{0.17em}\pm 1\hspace{0.17em}\text{sequence for}\hspace{0.17em}\lambda )}\lambda ,& D\lambda =q^{\left(\#\; of\; boxes\; of\; content\hspace{0.17em}0\hspace{0.17em}\text{in}\hspace{0.17em}\lambda \right)}\lambda \text{.}\end{array}$$

(a)

These formulas make ${ℱ}_{\mu }$ into a $U_v{\widehat{𝔰𝔩}}_{\ell }\text{-module}$

(b)

If $L_{\mu }=\mathbb{Z}[q,q^{-1}]\text{-span}\{\text{multisegments}\hspace{0.17em}\lambda =\lambda /\mu \}$ so that the multisegments form a $\mathbb{Z}[q,q^{-1}]$ basis of $L_{\mu }$ then $${\stackrel{\sim }{e}}_i\left[\lambda \right]=\left[{\stackrel{\sim }{e}}_i\lambda \right]\hspace{0.17em}\text{mod}\hspace{0.17em}q{L}_{\mu }\text{and}{\stackrel{\sim }{f}}_i\left[\lambda \right]=\left[{\stackrel{\sim }{f}}_i\lambda \right]\hspace{0.17em}\text{mod}\hspace{0.17em}q{L}_{\mu }\text{.}$$

		Proof.
	The permutations of the sequence $+1$ $+1,\dots ,+1,-1,-1,\dots ,-1$ are indexed by the elements of $S_t/S_k\times S_{t-k}$ where $t$ is the number of nodes after $(-1,+1)$ pairing. The group ${(\mathbb{Z}/2\mathbb{Z})}^p$ acts on the $(-1,+1)$ pairs by changing a pair $(-1,+1)$ to $(+1,-1)\text{.}$ For each $1\le k\le r$ define $$u_k=\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}\left(\sigma \tau \lambda \left[k\right]\right)\text{.}$$ Then $$u_k=\lambda \left[\lambda \right]\hspace{0.17em}\text{mod}\hspace{0.17em}q{L}_{\mu }\text{and}{e}_i{u}_k=\left[k\right]u_{k-1}\text{.}$$ The first statement is clear. To obtain the second statement $$\begin{array}{ccc}{e}_i{u}_k& =& \sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}\left(e_i\sigma \tau \lambda \left[k\right]\right)\\ & =& \sum _{e_i\hspace{0.17em}\text{changes a pair}}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & & +\sum _{e_i\hspace{0.17em}\text{changes a node}}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & =& 0+\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{e_i\hspace{0.17em}\text{changes a node}}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & =& \sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}\left[k\right]\left(\sum _{\sigma \in S_t/S_{k-1}\times S_{t-k+1}}{q}^{\ell \left(\sigma \right)}{(-1)}^{\ell \left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\right)\end{array}$$ $\square$

The Hall algebra

A quiver $(I,{\Omega }^{+})$ is a directed graph with vertex set $I$ and edge set ${\Omega }^{+}\subseteq I\times I,$ with no loops. A nilpotent representation of the quiver $(I,{\Omega }^{+})$ is a pair $(V,x)$ consisting of an $I\text{-graded}$ vector space over ${𝔽}_q,$ $$V=\underset{i\in I}{⨁}{V}_i\text{and an element}x\in \underset{(i\to j)\in {\Omega }^{+}}{⨁}\text{Hom}(V_i,V_j),$$ which is nilpotent as an element of $\text{End}\left(V\right)\text{.}$ The dimension of $V$ is the vector $$\text{dim}\left(V\right)={\left(\text{dim}\left(V_i\right)\right)}_{i\in I}$$ in ${\left({\mathbb{Z}}_{\ge 0}\right)}^I\text{.}$ A morphism $\varphi \in \text{Hom}(V,W)$ is a map $$\varphi \in \underset{i\in I}{⨁}\text{Hom}(V_i,W_i)\text{such that}{\varphi }_j{x}_{i\to j}=x_{i\to j}{\varphi }_i,$$ for all edges $i\to j\in {\Omega }^{+}\text{.}$ An extension $F\in {\text{Ext}}^1(V,W)$ is an exact sequence $$0⟶V⟶F⟶W⟶0$$ of morphisms of representations.

The following proposition shows that the constants $\text{dim}\hspace{0.17em}\text{Hom}(M,N)$ and $\text{dim}\hspace{0.17em}{\text{Ext}}^1(M,N)$ depend only on the dimension vectors of $M$ and $N\text{.}$

Let $M$ and $N$ be representations of $(I,{\Omega }^{+})$ with dimension vectors $\mu$ and $\nu$ respectively.

(a)	$\text{dim}\hspace{0.17em}\text{Hom}(M,N)=\text{???},$
(b)	$\text{dim}\hspace{0.17em}{\text{Ext}}^1(M,N)=\text{???}$
(c)	${\text{Ext}}^i(M,N)=0,$ for all $i>1\text{.}$
(d)	$\chi (M,N)=\text{dim}\left(\text{Hom}(M,N)\right)-\text{dim}\left({\text{Ext}}^1(M,N)\right)=-{\displaystyle \sum _{i\to j\in {\Omega }^{+}}}{\mu }_i{\nu }_j+{\displaystyle \sum _{i\in I}}{\mu }_i{\nu }_i\text{.}$

		Proof.
	$\square$

The Ringel-Hall algebra is the vector space ${𝒰}^{-}$ with basis $$\left\{E_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}{E}_{\lambda }\hspace{0.17em}\text{is an isomorphism class of nilpotent representations of}\hspace{0.17em}(I,{\Omega }^{+})\}$$ with multiplication given by $$E_{\mu }{E}_{\nu }=\sum _{\lambda }{q}^{-\text{dim}\hspace{0.17em}\text{Hom}(\mu ,\mu )-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,\nu )+\text{dim}\hspace{0.17em}\text{Hom}(\lambda ,\lambda )-\chi (\mu ,\nu )}{F}_{\mu \nu }^{\lambda }\left(q^{-2}\right)E_{\lambda },$$ where $$\begin{array}{ccc}{F}_{\mu \nu }^{\lambda }\left(q\right)& =& \text{Card}\{\gamma \subseteq \lambda \hspace{0.17em}|\hspace{0.17em}\gamma \cong \nu \hspace{0.17em}\text{and}\hspace{0.17em}\lambda /\gamma \cong \mu \}\\ & =& \left(\text{\# of submodules}\hspace{0.17em}\gamma \hspace{0.17em}\text{of}\hspace{0.17em}\lambda \hspace{0.17em}\text{of type}\hspace{0.17em}\nu \hspace{0.17em}\text{and cotype}\hspace{0.17em}\nu \right)\text{.}\end{array}$$

The type A case

The indecomposable representations $${[i,\ell )}_k=\{\begin{array}{cc}{𝔽}_q,& \text{if}\hspace{0.17em}i\le k\le i+\ell -1,\\ 0,& \text{otherwise,}\end{array}$$ of $(I,{\Omega }^{+})$ are identified with segments. By the analogue of the Krull-Schmidt theorem for representations of quivers every representation is isomorphic to a direct sum of indecomposable representations and so the isomorphism classes of representations of $(I,{\Omega }^{+})$ are identified with multisegments. Let $E_{\nu }$ denote the isomorphism class of the representation $\nu$ and let $E_i=E_{[i,1)}\text{.}$ Then $$\left\{E_{\nu }\hspace{0.17em}\right|\hspace{0.17em}\nu \hspace{0.17em}\text{is a multisegment}\}\text{is a basis of the Hall algebra}{𝒰}^{-}\text{.}$$

Let $M$ and $N$ be representations of $(I,{\Omega }^{+})$ with dimension vectors $\mu$ and $\nu$ respectively.

(a)	$\chi (M,N)=\text{dim}\left(\text{Hom}(M,N)\right)-\text{dim}\left({\text{Ext}}^1(M,N)\right)=-{\displaystyle \sum _{i\to j\in {\Omega }^{+}}}{\mu }_i{\nu }_j+{\displaystyle \sum _{i\in I}}{\mu }_i{\nu }_i\text{.}$
(b)	$\text{dim}\hspace{0.17em}\text{Hom}(E_{[j,\ell ]},E_{[i,k]})=\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}i\le j\le k\le \ell ,\\ 0,& \text{otherwise.}\end{array}$
(c)	$\text{dim}\hspace{0.17em}{\text{Ext}}^1(E_{[i,k]},E_{[j,\ell ]})=\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}i\le j\le k\le \ell ,\\ 0,& \text{otherwise.}\end{array}$

		Proof.
	(a) Since $\chi (M,N)$ is linear in $M$ and linear in $N$ it is sufficient to check that the formula is correct on indecomposable modules. This follows directly from parts (b) and (c). (b) Let $\{m_i,m_{i+1},\cdots ,m_j\}$ be a basis of $M$ such that $x_{i\to (i+1)}\left(m_i\right)=m_{i+1}$ and let $\{n_i,\dots ,n_k\}$ be a basis of $N$ such that $y_{r\to (r+1)}\left(n_r\right)=n_{r+1}\text{.}$ Any homomorphism $\varphi \in \text{Hom}(M,N)$ must have $\text{ker}\hspace{0.17em}\varphi$ being a submodule of $M$ and $\text{im}\hspace{0.17em}\varphi$ being a submodule of $N$ and so the only elements of $\text{Hom}(M,N)$ are multiples of the map $\varphi :M\to N$ given by $$\varphi \left(m_r\right)=\{\begin{array}{cc}{n}_r,& \text{if}\hspace{0.17em}j\le r\le \ell \hspace{0.17em}\text{and}\hspace{0.17em}i\le r\le k,\\ 0,& \text{otherwise.}\end{array}$$ $\square$

(a)	$F_{\nu ,i}^{{\nu }^{+}}\left(q\right)=\{\begin{array}{cc}{q}^{{\nu }^{+}(>\ell ;i]}(1+q+\cdots +q^{{\nu }^{+}(\ell ;i]-i}),& \text{if}\hspace{0.17em}{\nu }^{+}=\nu -(\ell -1;i-1]+(\ell ;i],\\ 0,& \text{otherwise.}\end{array}$
(b)	$F_{i,\nu }^{{}^{+}\nu }\left(q\right)=\{\begin{array}{cc}{q}^{{}^{+}\nu (i;l\ell ]}(1+q+\cdots +q^{{}^{+}\nu [i,\ell )-1}),& \text{if}\hspace{0.17em}{}^{+}\nu =\nu -[i-1;\ell -1)+[i;\ell ),\\ 0,& \text{otherwise.}\end{array}$

		Proof.
	(a) Count submodules of type $i$ in ${\nu }^{+}$ such that the quotient is of type $\nu \text{.}$ These are $1\text{-dimensional}$ spaces $P$ in $\nu {\left((\ge \ell ;i]\right)}_i$ which are not completely contained in $\nu {\left((>\ell ;i]\right)}_i,$ i.e. $P$ satisfies (a) $\text{dim}\left(P\right)=1,$ (b) $P\subseteq {\nu }_i^{+},$ (c) $P\cap {\nu }^{+}{\left((<\ell ;i]\right)}_i=0,$ (d) $P⊈{\nu }^{+}{\left((>\ell ,i]\right)}_i\text{.}$ The number of such subspaces is $$\frac{q^{\text{\#}(\ge \ell ;i]}-1}{q-1}-\frac{q^{\text{\#}(>\ell ;i]}-1}{q-1}=q^{\text{\#}(>\ell ;i]}\left(\frac{q^{{\nu }^{+}\left((\ell ;i]\right)}-1}{q-1}\right)\text{.}$$ (b) Count submodules of ${\nu }^{+}$ of type $\nu$ such that the quotient is of type $i\text{.}$ Choosing such a submodule amounts to choosing a codimension $1$ space in the $i\text{th}$ graded part of $[i;\le \ell )$ which is not completely contained in $[i;<\ell )\text{.}$ The number of such subspaces is $$\begin{array}{c}\frac{(q^{\text{\#}[i;\le \ell )}-1)(q^{\text{\#}[i;\le \ell )}-q)\cdots (q^{\text{\#}[i;\le \ell )}-q^{\text{\#}[i;\le \ell )-2})}{(q^{\text{\#}[i;\le \ell )-1}-1)(q^{\text{\#}[i;\le \ell )-1}-q)\cdots (q^{\text{\#}[i;\le \ell )-1}-q^{\text{\#}[i;\le \ell )-2})}\\ -\frac{(q^{\text{\#}[i;<\ell )}-1)(q^{\text{\#}[i;<\ell )}-q)\cdots (q^{\text{\#}[i;<\ell )}-q^{\text{\#}[i;<\ell )-2})}{(q^{\text{\#}[i;<\ell )-1}-1)(q^{\text{\#}[i;<\ell )-1}-q)\cdots (q^{\text{\#}[i;<\ell )-1}-q^{\text{\#}[i;<\ell )-2})}\\ =\frac{(q^{\text{\#}[i;\le \ell )}-1)}{q-1}-\frac{(q^{\text{\#}[i;<\ell )}-1)}{q-1}=q^{\text{\#}[i;<\ell )}\frac{(q^{\text{\#}[i;\ell )}-1)}{q-1}\text{.}\end{array}$$ $\square$

For each multisegment $$\nu =\left(\begin{array}{ccccccc}{\lambda }_1& & {\lambda }_2& & \cdots & & {\lambda }_r\\ d_1& \ge & d_2& \ge & \cdots & \ge & d_r\end{array}\right]\text{place}\begin{array}{c}+1\hspace{0.17em}\text{over each}\hspace{0.17em}{\lambda }_j=i-1,\\ -1\hspace{0.17em}\text{over each}\hspace{0.17em}{\lambda }_j=i,\\ 0\hspace{0.17em}\text{over each}\hspace{0.17em}{\lambda }_j\ne =i,i-1\text{.}\end{array}$$

Let $\left[M\right]$ be the class of the representation indexed by the multisegment $M\text{.}$ $$\begin{array}{ccc}{E}_i{E}_{\nu }& =& \sum _{c({\nu }^{+}/\nu )=i}{q}^{(\text{sum of the}\hspace{0.17em}\pm 1\hspace{0.17em}\text{before}\hspace{0.17em}{\nu }^{+}/\nu )}{E}_{{\nu }^{+}}\text{and}\\ \left[M\right]\left[e_i\right]& =& \sum _{[M-]}{q}^{(\text{sum of the labels after}\hspace{0.17em}{}^{+}M/M)}\left[{}^{+}M\right],\end{array}$$ where the first sum is over all multisegments ${\nu }^{+}$ which are obtained from $\nu$ by adding a box of content $i$ to the end of a row of $\nu ,$ and the second sum is over all multisegments ${}^{+}M$ obtained from $M$ by adding a box of content $i$ to the beginning of a row of $M\text{.}$

		Proof.
	(a) Let ${\nu }^{+}=\nu +(\ell ,i]-(\ell -1;i-1]\text{.}$ Then $$\begin{array}{c}\text{dim}\hspace{0.17em}\text{Hom}({\nu }^{+},{\nu }^{+})-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,\nu )\\ =(\text{dim}\hspace{0.17em}\text{Hom}(\nu ,(\ell ,i])-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,(\ell -1,i-1]))\\ +(\text{dim}\hspace{0.17em}\text{Hom}((\ell ,i],\nu )-\text{dim}\hspace{0.17em}\text{Hom}((\ell -1,i-1],\nu ))\\ +\text{dim}\hspace{0.17em}\text{Hom}((\ell ,i],(\ell ,i])-\text{dim}\hspace{0.17em}\text{Hom}((\ell ,i],(\ell -1,i-1])\\ +\text{dim}\hspace{0.17em}\text{Hom}((\ell -1,i-1],(\ell ,i])+\text{dim}\hspace{0.17em}\text{Hom}((\ell -1,i-1],(\ell -1,i-1])\\ =(\nu [i,>0)-\nu (\le \ell -1,i-1])+(\nu (\ge \ell ,i]-0)+1-1-0+1\\ =\nu [i,>0)-\nu (\le \ell -1,i-1]+\nu (\ge \ell ,i]+1\text{.}\end{array}$$ Since $\text{dim}\hspace{0.17em}\text{Hom}(i,i)=1$ and $\chi (\nu ,i)=-{\nu }_{i-1}+{\nu }_i=\nu [i,>0)-\nu (>0,i-1],$ $$-\text{dim}\hspace{0.17em}\text{Hom}(i,i)-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,\nu )+\text{dim}\hspace{0.17em}\text{Hom}({\nu }^{+},{\nu }^{+})-\chi (\nu ,i)=\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]\text{.}$$ Thus the coefficient of $M_{{\nu }^{+}}$ in $E_{\nu }{E}_i$ is $$\begin{array}{ccc}{q}^{\nu (\ge \ell ,i-1]+\nu (\ge \ell ,i]}{F}_{\nu ,i}^{{\nu }^{+}}\left(q^{-2}\right)& =& q^{\nu (\ge \ell ,i-1]+\nu (\ge \ell mi]}{q}^{-2{\nu }^{+}(>\ell ;i]}(1+q^{-2}+\cdots +q^{-2{\nu }^{+}(\ell ,i]+2})\\ & =& q^{\nu (\ge \ell ,i-1]+\nu (\ge \ell mi]}{q}^{-2\nu (>\ell ;i]}{q}^{-2(\nu (\ell ,i]+1)}{q}^2(1+q^2+\cdots +q^{2\nu (\ell ,i]})\\ & =& q^{\nu (\ge \ell ,i-1]-\nu (\ge \ell mi]}(1+q^2+\cdots +q^{2\nu (\ell ,i]})\text{.}\end{array}$$ (b) Let ${}^{+}\nu =\nu +[i,\ell )-[i+1,\ell -1)\text{.}$ Then $$\begin{array}{c}\text{dim}\hspace{0.17em}\text{Hom}({}^{+}\nu ,{}^{+}\nu )-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,\nu )\\ =(\text{dim}\hspace{0.17em}\text{Hom}(\nu ,[i,\ell ))-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,[i+1,\ell -1)))\\ +(\text{dim}\hspace{0.17em}\text{hom}([i,\ell ),\nu )-\text{dim}\hspace{0.17em}\text{Hom}([i+1,\ell -1),\nu ))\\ +\text{dim}\hspace{0.17em}\text{Hom}([i,\ell ),[i,\ell ))-\text{dim}\hspace{0.17em}\text{Hom}([i,\ell ),[i+1,\ell -1))\\ +\text{dim}\hspace{0.17em}\text{Hom}([i+1,\ell -1),[i,\ell ))+\text{dim}\hspace{0.17em}\text{Hom}([i+1,\ell -1),[i+1,\ell -1))\\ =(\nu [i,\ge \ell )-0)+(\nu (>0,i]-\nu [i+1,\le \ell -1))+1-0-1+1\\ =\nu (>0,i]+\nu [i,\ge \ell )-\nu [i+1,\le i-1)+1\text{.}\end{array}$$ Since $\text{dim}\hspace{0.17em}\text{Hom}(i,i)=1$ and $\chi (i,\nu )=-{\nu }_{i+1}+{\nu }_i=\nu (>0,i]-\nu [i+1,>0),$ $$-\text{dim}\hspace{0.17em}\text{Hom}(i,i)-\text{dim}\hspace{0.17em}\text{Hom}(\nu ,\nu )+\text{dim}\hspace{0.17em}\text{Hom}({}^{+}\nu ,{}^{+}\nu )-\chi (i,\nu )=\nu [i,\ge \ell )+\nu [i+1,\ge \ell )\text{.}$$ Thus the coefficient of $M_{{}^{+}\nu }$ in $E_i{E}_{\nu }$ is $$\begin{array}{ccc}{q}^{\nu [i,\ge \ell )+\nu [i+1,\ge \ell )}{F}_{i,\nu }^{{}^{+}\nu }\left(q^{-2}\right)& =& q^{\nu [i,\ge \ell )+\nu [i+1,\ge \ell )}{q}^{-2{}^{+}\nu [i,>\ell )}(1+q^{-2}+\cdots +q^{-2{}^{+}\nu [i,\ell )+2})\\ & =& q^{\nu [i,\ge \ell )+\nu [i+1,\ge \ell )}{q}^{-2\nu [i,>\ell )}{q}^{-2(\nu [i,\ell )+1)}{q}^2(1+q^2+\cdots +q^{2\nu [i,\ell )})\\ & =& q^{\nu [i+1,\ge \ell )-\nu [i,\ge \ell )}(1+q^2+\cdots +q^{2\nu [i,\ell )})\text{.}\end{array}$$ $\square$

For each vertex $i\in I$ let $\left[e_i\right]$ be the class of the representation given by $$V_i=\{\begin{array}{cc}{𝔽}_q,& \text{if}\hspace{0.17em}j=i,\\ 0,& \text{if}\hspace{0.17em}j\ne i,\end{array}$$ and, for each edge $i\to j$ in ${\Omega }^{+}$ let $e_{ij}$ be the representation given by $$V_k=\{\begin{array}{cc}{𝔽}_q,& \text{if}\hspace{0.17em}k=i\hspace{0.17em}\text{or}\hspace{0.17em}k=j,\\ 0,& \text{otherwise}\end{array}\text{and}{x}_{ij}={\text{id}}_{{𝔽}_q}\text{.}$$

Let $(I,{\Omega }^{+})$ be a type $A$ quiver with the canonical orientiation. Then $$\begin{array}{ccc}{E}_i^2{E}_{i+1}-(q+q^{-1})E_i{E}_{i+1}{E}_i+E_{i+1}{E}_i^2& =& 0\text{and}\\ E_{i+1}^2{E}_i-(q+q^{-1})E_i{E}_{i+1}{E}_i+E_i{E}_{i+1}^2& =& 0,\end{array}$$ in the Hall algebra.

		Proof.
	Using Proposition ??? we get $$\begin{array}{ccc}{E}_i^2& =& q^{-1}{E}_{2(1,i]},\\ E_{i+1}^2& =& q^{-2}{E}_{2(1,i+1]},\end{array}\text{and}\begin{array}{ccc}{E}_i{E}_{i+1}& =& q{E}_{(1,i+1]+(1,i]}+E_{(2,i+1]},\\ E_{i+1}{E}_i& =& E_{(1,i]+(1,i+1]},\end{array}$$ and $$\begin{array}{ccc}{E}_i^2{E}_{i+1}& =& q{E}_{(1,i+1]+2(1,i]}+q{E}_{(2,i+1]+(1,i]}+q^{-1}{E}_{(2,i+1]+(1,i]},\\ E_i{E}_{i+1}{E}_i& =& E_{(1,i+1]+2(1,i]}+E_{(2,i+1]+(1,i]},\\ E_i{E}_{i+1}^2& =& q^{-2}{E}_{(1,i+1]+2(1,i]}\text{.}\end{array}$$ The result follows. The calculation for the other case is similar. $\square$

Notes and References

[Am1] Y. Amit, Convergence properties of the Gibbs sampler for perturbations of Gaussians, Ann. Statistics 24 (1995), 122–140.

[CG] N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.

[KKM] S.-J. Kang, M. Kashiwara, and K. Misra, ?? Composition Math. 92 no. 3 (1994), 299-325.

$\text{[}{K}^2{M}^2{N}^2\text{]}$ S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, Nakyashiki, Nakashima, ?? Int. J. Mod. Phys. A 7 Suppl. 1A (1992), 449-484, Proc. RIMS Proj. 1991 Infinite Analysis.

[KS] M. Kashiwara, ?? Duke Math. J. 73 no. ? (1994), 383-413.

[KS] M. Kashiwara and Y. Saito ?? Duke Math. J. 89 no. 1 (1997), ???–???.

Notes and references

This is a typed exert of Representation theory Lecture notes: Chapter 6 by Arun Ram. Research supported in part by National Science Foundation grant DMS-9622985.

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